{"id":14216,"date":"2018-09-27T16:41:09","date_gmt":"2018-09-27T16:41:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/sum-to-product-and-product-to-sum-formulas\/"},"modified":"2020-05-21T05:10:10","modified_gmt":"2020-05-21T05:10:10","slug":"sum-to-product-and-product-to-sum-formulas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/chapter\/sum-to-product-and-product-to-sum-formulas\/","title":{"raw":"Section 5.4: Sum-to-Product and Product-to-Sum Formulas","rendered":"Section 5.4: Sum-to-Product and Product-to-Sum Formulas"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400\">Express products as sums.<\/li>\r\n \t<li style=\"font-weight: 400\">Express sums as products.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Expressing Products as Sums<\/h2>\r\nWe have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the <strong>product-to-sum formulas<\/strong>, which express products of trigonometric functions as sums. Let\u2019s investigate the cosine identity first and then the sine identity.\r\n<h3>Expressing Products as Sums for Cosine<\/h3>\r\nWe can derive the product-to-sum formula from the sum and difference identities for <strong>cosine<\/strong>. If we add the two equations, we get:\r\n<p style=\"text-align: center\">[latex]\\begin{gathered}\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta =\\cos \\left(\\alpha -\\beta \\right)\\\\\\underline{ +\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta =\\cos \\left(\\alpha +\\beta \\right)} \\\\ 2\\cos \\alpha \\cos \\beta =\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\end{gathered}[\/latex]<\/p>\r\n\r\n<div>Then, we divide by [latex]2[\/latex] to isolate the product of cosines:<\/div>\r\n<div style=\"text-align: center\">[latex]\\cos \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a product of cosines, express as a sum.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Write the formula for the product of cosines.<\/li>\r\n \t<li>Substitute the given angles into the formula.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Writing the Product as a Sum Using the Product-to-Sum Formula for Cosine<\/h3>\r\nWrite the following product of cosines as a sum: [latex]2\\cos \\left(\\frac{7x}{2}\\right)\\cos \\frac{3x}{2}[\/latex].\r\n\r\n[reveal-answer q=\"440362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"440362\"]\r\n\r\nWe begin by writing the formula for the product of cosines:\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\r\nWe can then substitute the given angles into the formula and simplify.\r\n<p style=\"text-align: center\">[latex]\\begin{align}2\\cos \\left(\\frac{7x}{2}\\right)\\cos \\left(\\frac{3x}{2}\\right)&amp;=\\left(2\\right)\\left(\\frac{1}{2}\\right)\\left[\\cos \\left(\\frac{7x}{2}-\\frac{3x}{2}\\right)+\\cos \\left(\\frac{7x}{2}+\\frac{3x}{2}\\right)\\right] \\\\ &amp;=\\left[\\cos \\left(\\frac{4x}{2}\\right)+\\cos \\left(\\frac{10x}{2}\\right)\\right] \\\\ &amp;=\\cos 2x+\\cos 5x \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nUse the product-to-sum formula to write the product as a sum or difference: [latex]\\cos \\left(2\\theta \\right)\\cos \\left(4\\theta \\right)[\/latex].\r\n\r\n[reveal-answer q=\"293192\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"293192\"][latex]\\frac{1}{2}\\left(\\cos 6\\theta +\\cos 2\\theta \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]4642[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Expressing the Product of Sine and Cosine as a Sum<\/h2>\r\nNext, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for <strong>sine<\/strong>. If we add the sum and difference identities, we get:\r\n<div style=\"text-align: center\">[latex]\\begin{gathered}\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta \\\\\\underline{ +\\text{ }\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta}\\\\ \\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)=2\\sin \\alpha \\cos \\beta \\end{gathered}[\/latex]<\/div>\r\nThen, we divide by 2 to isolate the product of cosine and sine:\r\n<div style=\"text-align: center\">[latex]\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Writing the Product as a Sum Containing only Sine or Cosine<\/h3>\r\nExpress the following product as a sum containing only sine or cosine and no products: [latex]\\sin \\left(4\\theta \\right)\\cos \\left(2\\theta \\right)[\/latex].\r\n\r\n[reveal-answer q=\"816809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"816809\"]\r\n\r\nWrite the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\sin \\alpha \\cos \\beta &amp;=\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right] \\\\ \\sin \\left(4\\theta \\right)\\cos \\left(2\\theta \\right)&amp;=\\frac{1}{2}\\left[\\sin \\left(4\\theta +2\\theta \\right)+\\sin \\left(4\\theta -2\\theta \\right)\\right] \\\\ &amp;=\\frac{1}{2}\\left[\\sin \\left(6\\theta \\right)+\\sin \\left(2\\theta \\right)\\right] \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nUse the product-to-sum formula to write the product as a sum: [latex]\\sin \\left(x+y\\right)\\cos \\left(x-y\\right)[\/latex].\r\n\r\n[reveal-answer q=\"276662\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"276662\"]\r\n\r\n[latex]\\frac{1}{2}\\left(\\sin 2x+\\sin 2y\\right)\\\\[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]4631[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Expressing Products of Sines in Terms of Cosine<\/h2>\r\nExpressing the product of sines in terms of <strong>cosine<\/strong> is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:\r\n<p style=\"text-align: center\">[latex]\\begin{gathered}\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta \\\\ \\underline{ -\\text{ }\\cos \\left(\\alpha +\\beta \\right)=-\\left(\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta \\right)} \\\\ \\cos \\left(\\alpha -\\beta \\right)-\\cos \\left(\\alpha +\\beta \\right)=2\\sin \\alpha \\sin \\beta \\end{gathered}[\/latex]<\/p>\r\nThen, we divide by 2 to isolate the product of sines:\r\n<div style=\"text-align: center\">[latex]\\sin \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)-\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/div>\r\nSimilarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Product-to-Sum Formulas<\/h3>\r\nThe <strong>product-to-sum formulas<\/strong> are as follows:\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\sin \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)-\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)-\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Express the Product as a Sum or Difference<\/h3>\r\nWrite [latex]\\cos \\left(3\\theta \\right)\\cos \\left(5\\theta \\right)[\/latex] as a sum or difference.\r\n\r\n[reveal-answer q=\"145283\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145283\"]\r\n\r\nWe have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and simplify.\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\cos \\alpha \\cos \\beta &amp;=\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right] \\\\ \\cos \\left(3\\theta \\right)\\cos \\left(5\\theta \\right)&amp;=\\frac{1}{2}\\left[\\cos \\left(3\\theta -5\\theta \\right)+\\cos \\left(3\\theta +5\\theta \\right)\\right] \\\\ &amp;=\\frac{1}{2}\\left[\\cos \\left(2\\theta \\right)+\\cos \\left(8\\theta \\right)\\right] &amp;&amp; \\text{Use even-odd identity}. \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nUse the product-to-sum formula to evaluate [latex]\\cos \\frac{11\\pi }{12}\\cos \\frac{\\pi }{12}[\/latex].\r\n\r\n[reveal-answer q=\"731318\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"731318\"]\r\n\r\n[latex]\\frac{-2-\\sqrt{3}}{4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Expressing Sums as Products<\/h2>\r\nSome problems require the reverse of the process we just used. The <strong>sum-to-product formulas<\/strong> allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for <strong>sine<\/strong>. Let [latex]\\frac{u+v}{2}=\\alpha [\/latex] and [latex]\\frac{u-v}{2}=\\beta [\/latex].\r\n\r\nThen,\r\n<div style=\"text-align: center\">[latex]\\begin{align}\\alpha +\\beta &amp;=\\frac{u+v}{2}+\\frac{u-v}{2} \\\\ &amp;=\\frac{2u}{2} \\\\ &amp;=u \\\\ \\text{ } \\\\ \\alpha -\\beta &amp;=\\frac{u+v}{2}-\\frac{u-v}{2} \\\\ &amp;=\\frac{2v}{2} \\\\ &amp;=v \\end{align}[\/latex]<\/div>\r\nThus, replacing [latex]\\alpha [\/latex] and [latex]\\beta [\/latex] in the product-to-sum formula with the substitute expressions, we have\r\n<div style=\"text-align: center\">[latex]\\begin{align}&amp;\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right] \\\\ &amp;\\sin \\left(\\frac{u+v}{2}\\right)\\cos \\left(\\frac{u-v}{2}\\right)=\\frac{1}{2}\\left[\\sin u+\\sin v\\right]&amp;&amp; \\text{Substitute for}\\left(\\alpha +\\beta \\right)\\text{ and }\\left(\\alpha -\\beta \\right) \\\\ &amp;2\\sin \\left(\\frac{u+v}{2}\\right)\\cos \\left(\\frac{u-v}{2}\\right)=\\sin u+\\sin v \\end{align}[\/latex]<\/div>\r\nThe other sum-to-product identities are derived similarly.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sum-to-Product Formulas<\/h3>\r\nThe <strong>sum-to-product formulas<\/strong> are as follows:\r\n<p style=\"text-align: center\">[latex]\\sin \\alpha +\\sin \\beta =2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\sin \\alpha -\\sin \\beta =2\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha -\\cos \\beta =-2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha +\\cos \\beta =2\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 4: Writing the Difference of Sines as a Product<\/h3>\r\nWrite the following difference of sines expression as a product: [latex]\\sin \\left(4\\theta \\right)-\\sin \\left(2\\theta \\right)[\/latex].\r\n\r\n[reveal-answer q=\"627723\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"627723\"]\r\n\r\nWe begin by writing the formula for the difference of sines.\r\n<p style=\"text-align: center\">[latex]\\sin \\alpha -\\sin \\beta =2\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)[\/latex]<\/p>\r\nSubstitute the values into the formula, and simplify.\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\sin \\left(4\\theta \\right)-\\sin \\left(2\\theta \\right)&amp;=2\\sin \\left(\\frac{4\\theta -2\\theta }{2}\\right)\\cos \\left(\\frac{4\\theta +2\\theta }{2}\\right) \\\\ &amp;=2\\sin \\left(\\frac{2\\theta }{2}\\right)\\cos \\left(\\frac{6\\theta }{2}\\right) \\\\ &amp;=2\\sin \\theta \\cos \\left(3\\theta \\right) \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nUse the sum-to-product formula to write the sum as a product: [latex]\\sin \\left(3\\theta \\right)+\\sin \\left(\\theta \\right)[\/latex].\r\n\r\n[reveal-answer q=\"34084\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"34084\"]\r\n\r\n[latex]2\\sin \\left(2\\theta \\right)\\cos \\left(\\theta \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]4632[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 5: Evaluating Using the Sum-to-Product Formula<\/h3>\r\nEvaluate [latex]\\cos \\left({15}^{\\circ }\\right)-\\cos \\left({75}^{\\circ }\\right)[\/latex].\r\n\r\n[reveal-answer q=\"296492\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"296492\"]\r\n\r\nWe begin by writing the formula for the difference of cosines.\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha -\\cos \\beta =-2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\r\nThen we substitute the given angles and simplify.\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\cos \\left({15}^{\\circ }\\right)-\\cos \\left({75}^{\\circ }\\right)&amp;=-2\\sin \\left(\\frac{{15}^{\\circ }+{75}^{\\circ }}{2}\\right)\\sin \\left(\\frac{{15}^{\\circ }-{75}^{\\circ }}{2}\\right) \\\\ &amp;=-2\\sin \\left({45}^{\\circ }\\right)\\sin \\left(-{30}^{\\circ }\\right) \\\\ &amp;=-2\\left(\\frac{\\sqrt{2}}{2}\\right)\\left(-\\frac{1}{2}\\right) \\\\ &amp;=\\frac{\\sqrt{2}}{2}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 6: Proving an Identity<\/h3>\r\nProve the identity:\r\n<p style=\"text-align: center\">[latex]\\dfrac{\\cos \\left(4t\\right)-\\cos \\left(2t\\right)}{\\sin \\left(4t\\right)+\\sin \\left(2t\\right)}=-\\tan t[\/latex]<\/p>\r\n[reveal-answer q=\"514870\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"514870\"]\r\n\r\nWe will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\frac{\\cos \\left(4t\\right)-\\cos \\left(2t\\right)}{\\sin \\left(4t\\right)+\\sin \\left(2t\\right)}&amp;=\\frac{-2\\sin \\left(\\frac{4t+2t}{2}\\right)\\sin \\left(\\frac{4t - 2t}{2}\\right)}{2\\sin \\left(\\frac{4t+2t}{2}\\right)\\cos \\left(\\frac{4t - 2t}{2}\\right)} \\\\ &amp;=\\frac{-2\\sin \\left(3t\\right)\\sin t}{2\\sin \\left(3t\\right)\\cos t} \\\\ &amp;=\\frac{-2\\sin \\left(3t\\right)\\sin t}{2\\sin \\left(3t\\right)\\cos t} \\\\ &amp;=-\\frac{\\sin t}{\\cos t} \\\\ &amp;=-\\tan t \\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nRecall that verifying trigonometric identities has its own set of rules. The procedures for solving an equation are not the same as the procedures for verifying an identity. When we prove an identity, we pick one side to work on and make substitutions until that side is transformed into the other side.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 7: Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities<\/h3>\r\nVerify the identity [latex]{\\csc }^{2}\\theta -2=\\dfrac{\\cos \\left(2\\theta \\right)}{{\\sin }^{2}\\theta }[\/latex].\r\n\r\n[reveal-answer q=\"225762\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"225762\"]\r\n\r\nFor verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\frac{\\cos \\left(2\\theta \\right)}{{\\sin }^{2}\\theta }&amp;=\\frac{1 - 2{\\sin }^{2}\\theta }{{\\sin }^{2}\\theta } \\\\ &amp;=\\frac{1}{{\\sin }^{2}\\theta }-\\frac{2{\\sin }^{2}\\theta }{{\\sin }^{2}\\theta } \\\\ &amp;={\\csc }^{2}\\theta -2\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nVerify the identity [latex]\\tan \\theta \\cot \\theta -{\\cos }^{2}\\theta ={\\sin }^{2}\\theta [\/latex].\r\n\r\n[reveal-answer q=\"591097\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"591097\"]\r\n\r\n[latex]\\begin{align}\\tan \\theta \\cot \\theta -{\\cos }^{2}\\theta &amp;=\\left(\\frac{\\sin \\theta }{\\cos \\theta }\\right)\\left(\\frac{\\cos \\theta }{\\sin \\theta }\\right)-{\\cos }^{2}\\theta \\\\ &amp;=1-{\\cos }^{2}\\theta \\\\ &amp;={\\sin }^{2}\\theta \\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<table id=\"fs-id1837805\" summary=\"..\"><colgroup> <col \/> <col \/> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>Product-to-sum Formulas<\/strong><\/td>\r\n<td>\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\sin \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)-\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)-\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Sum-to-product Formulas<\/strong><\/td>\r\n<td>\r\n<p style=\"text-align: center\">[latex]\\sin \\alpha +\\sin \\beta =2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\sin \\alpha -\\sin \\beta =2\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha -\\cos \\beta =-2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\cos \\alpha +\\cos \\beta =2\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.<\/li>\r\n \t<li>We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines.<\/li>\r\n \t<li>We can also derive the sum-to-product identities from the product-to-sum identities using substitution.<\/li>\r\n \t<li>We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines.<\/li>\r\n \t<li>Trigonometric expressions are often simpler to evaluate using the formulas.<\/li>\r\n \t<li>The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side.<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id2872047\" class=\"definition\">\r\n \t<dt>product-to-sum formula<\/dt>\r\n \t<dd id=\"fs-id2872050\">a trigonometric identity that allows the writing of a product of trigonometric functions as a sum or difference of trigonometric functions<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1548089\" class=\"definition\">\r\n \t<dt>sum-to-product formula<\/dt>\r\n \t<dd id=\"fs-id1548092\">a trigonometric identity that allows, by using substitution, the writing of a sum of trigonometric functions as a product of trigonometric functions<\/dd>\r\n<\/dl>\r\n<\/div>\r\n&nbsp;\r\n<h2 style=\"text-align: center\">Section 5.4 Homework Exercises<\/h2>\r\n1. Starting with the product to sum formula [latex]\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex], explain how to determine the formula for [latex]\\cos \\alpha \\sin \\beta [\/latex].\r\n\r\n2. Explain two different methods of calculating [latex]\\cos \\left(195^\\circ \\right)\\cos \\left(105^\\circ \\right)[\/latex], one of which uses the product to sum. Which method is easier?\r\n\r\n3. Explain a situation where we would convert an equation from a sum to a product and give an example.\r\n\r\n4.\u00a0Explain a situation where we would convert an equation from a product to a sum, and give an example.\r\n\r\nFor the following exercises, rewrite the product as a sum or difference.\r\n\r\n5. [latex]16\\sin \\left(16x\\right)\\sin \\left(11x\\right)[\/latex]\r\n\r\n6.\u00a0[latex]20\\cos \\left(36t\\right)\\cos \\left(6t\\right)[\/latex]\r\n\r\n7. [latex]2\\sin \\left(5x\\right)\\cos \\left(3x\\right)[\/latex]\r\n\r\n8.\u00a0[latex]10\\cos \\left(5x\\right)\\sin \\left(10x\\right)[\/latex]\r\n\r\n9. [latex]\\sin \\left(-x\\right)\\sin \\left(5x\\right)[\/latex]\r\n\r\n10.\u00a0[latex]\\sin \\left(3x\\right)\\cos \\left(5x\\right)[\/latex]\r\n\r\nFor the following exercises, rewrite the sum or difference as a product.\r\n\r\n11. [latex]\\cos \\left(6t\\right)+\\cos \\left(4t\\right)[\/latex]\r\n\r\n12.\u00a0[latex]\\sin \\left(3x\\right)+\\sin \\left(7x\\right)[\/latex]\r\n\r\n13. [latex]\\cos \\left(7x\\right)+\\cos \\left(-7x\\right)[\/latex]\r\n\r\n14.\u00a0[latex]\\sin \\left(3x\\right)-\\sin \\left(-3x\\right)[\/latex]\r\n\r\n15. [latex]\\cos \\left(3x\\right)+\\cos \\left(9x\\right)[\/latex]\r\n\r\n16.\u00a0[latex]\\sin h-\\sin \\left(3h\\right)[\/latex]\r\n\r\nFor the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.\r\n\r\n17. [latex]\\cos \\left(45^\\circ \\right)\\cos \\left(15^\\circ \\right)[\/latex]\r\n\r\n18.\u00a0[latex]\\cos \\left(45^\\circ \\right)\\sin \\left(15^\\circ \\right)[\/latex]\r\n\r\n19. [latex]\\sin \\left(-345^\\circ \\right)\\sin \\left(-15^\\circ \\right)[\/latex]\r\n\r\n20.\u00a0[latex]\\sin \\left(195^\\circ \\right)\\cos \\left(15^\\circ \\right)[\/latex]\r\n\r\n21. [latex]\\sin \\left(-45^\\circ \\right)\\sin \\left(-15^\\circ \\right)[\/latex]\r\n\r\nFor the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.\r\n\r\n22. [latex]\\cos \\left(23^\\circ \\right)\\sin \\left(17^\\circ \\right)[\/latex]\r\n\r\n23. [latex]2\\sin \\left(100^\\circ \\right)\\sin \\left(20^\\circ \\right)[\/latex]\r\n\r\n24.\u00a0[latex]2\\sin \\left(-100^\\circ \\right)\\sin \\left(-20^\\circ \\right)[\/latex]\r\n\r\n25. [latex]\\sin \\left(213^\\circ \\right)\\cos \\left(8^\\circ \\right)[\/latex]\r\n\r\n26.\u00a0[latex]2\\cos \\left(56^\\circ \\right)\\cos \\left(47^\\circ \\right)[\/latex]\r\n\r\nFor the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.\r\n\r\n27. [latex]\\sin \\left(76^\\circ \\right)+\\sin \\left(14^\\circ \\right)[\/latex]\r\n\r\n28.\u00a0[latex]\\cos \\left(58^\\circ \\right)-\\cos \\left(12^\\circ \\right)[\/latex]\r\n\r\n29. [latex]\\sin \\left(101^\\circ \\right)-\\sin \\left(32^\\circ \\right)[\/latex]\r\n\r\n30.\u00a0[latex]\\cos \\left(100^\\circ \\right)+\\cos \\left(200^\\circ \\right)[\/latex]\r\n\r\n31. [latex]\\sin \\left(-1^\\circ \\right)+\\sin \\left(-2^\\circ \\right)[\/latex]\r\n\r\nFor the following exercises, prove the identity.\r\n\r\n32. [latex]\\frac{\\cos \\left(a+b\\right)}{\\cos \\left(a-b\\right)}=\\frac{1-\\tan a\\tan b}{1+\\tan a\\tan b}[\/latex]\r\n\r\n33. [latex]4\\sin \\left(3x\\right)\\cos \\left(4x\\right)=2\\sin \\left(7x\\right)-2\\sin x[\/latex]\r\n\r\n34.\u00a0[latex]\\frac{6\\cos \\left(8x\\right)\\sin \\left(2x\\right)}{\\sin \\left(-6x\\right)}=-3\\sin \\left(10x\\right)\\csc \\left(6x\\right)+3[\/latex]\r\n\r\n35. [latex]\\sin x+\\sin \\left(3x\\right)=4\\sin x{\\cos }^{2}x[\/latex]\r\n\r\n36.\u00a0[latex]2\\left({\\cos }^{3}x-\\cos x{\\sin }^{2}x\\right)=\\cos \\left(3x\\right)+\\cos x[\/latex]\r\n\r\n37. [latex]2\\tan x\\cos \\left(3x\\right)=\\sec x\\left(\\sin \\left(4x\\right)-\\sin \\left(2x\\right)\\right)[\/latex]\r\n\r\n38. [latex]\\cos \\left(a+b\\right)+\\cos \\left(a-b\\right)=2\\cos a\\cos b[\/latex]\r\n\r\nFor the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.\r\n\r\n39. [latex]\\cos \\left({58}^{\\circ }\\right)+\\cos \\left({12}^{\\circ }\\right)[\/latex]\r\n\r\n40.\u00a0[latex]\\sin \\left({2}^{\\circ }\\right)-\\sin \\left({3}^{\\circ }\\right)[\/latex]\r\n\r\n41. [latex]\\cos \\left({44}^{\\circ }\\right)-\\cos \\left({22}^{\\circ }\\right)[\/latex]\r\n\r\n42.\u00a0[latex]\\cos \\left({176}^{\\circ }\\right)\\sin \\left({9}^{\\circ }\\right)[\/latex]\r\n\r\n43. [latex]\\sin \\left(-{14}^{\\circ }\\right)\\sin \\left({85}^{\\circ }\\right)[\/latex]\r\n\r\nFor the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.\r\n\r\n44. [latex]2\\sin \\left(2x\\right)\\sin \\left(3x\\right)=\\cos x-\\cos \\left(5x\\right)[\/latex]\r\n\r\n45. [latex]\\frac{\\cos \\left(10\\theta \\right)+\\cos \\left(6\\theta \\right)}{\\cos \\left(6\\theta \\right)-\\cos \\left(10\\theta \\right)}=\\cot \\left(2\\theta \\right)\\cot \\left(8\\theta \\right)[\/latex]\r\n\r\n46.\u00a0[latex]\\frac{\\sin \\left(3x\\right)-\\sin \\left(5x\\right)}{\\cos \\left(3x\\right)+\\cos \\left(5x\\right)}=\\tan x[\/latex]\r\n\r\n47. [latex]2\\cos \\left(2x\\right)\\cos x+\\sin \\left(2x\\right)\\sin x=2\\sin x[\/latex]\r\n\r\n48.\u00a0[latex]\\frac{\\sin \\left(2x\\right)+\\sin \\left(4x\\right)}{\\sin \\left(2x\\right)-\\sin \\left(4x\\right)}=-\\tan \\left(3x\\right)\\cot x[\/latex]\r\n\r\nFor the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.\r\n\r\n49. [latex]\\frac{\\sin \\left(9t\\right)-\\sin \\left(3t\\right)}{\\cos \\left(9t\\right)+\\cos \\left(3t\\right)}[\/latex]\r\n\r\n50.\u00a0[latex]2\\sin \\left(8x\\right)\\cos \\left(6x\\right)-\\sin \\left(2x\\right)[\/latex]\r\n\r\n51. [latex]\\frac{\\sin \\left(3x\\right)-\\sin x}{\\sin x}[\/latex]\r\n\r\n52.\u00a0[latex]\\frac{\\cos \\left(5x\\right)+\\cos \\left(3x\\right)}{\\sin \\left(5x\\right)+\\sin \\left(3x\\right)}[\/latex]\r\n\r\n53. [latex]\\sin x\\cos \\left(15x\\right)-\\cos x\\sin \\left(15x\\right)[\/latex]\r\n\r\nFor the following exercises, prove the following sum-to-product formulas.\r\n\r\n54. [latex]\\sin x-\\sin y=2\\sin \\left(\\frac{x-y}{2}\\right)\\cos \\left(\\frac{x+y}{2}\\right)[\/latex]\r\n\r\n55. [latex]\\cos x+\\cos y=2\\cos \\left(\\frac{x+y}{2}\\right)\\cos \\left(\\frac{x-y}{2}\\right)[\/latex]\r\n\r\nFor the following exercises, prove the identity.\r\n\r\n56. [latex]\\frac{\\sin \\left(6x\\right)+\\sin \\left(4x\\right)}{\\sin \\left(6x\\right)-\\sin \\left(4x\\right)}=\\tan \\left(5x\\right)\\cot x[\/latex]\r\n\r\n57. [latex]\\frac{\\cos \\left(3x\\right)+\\cos x}{\\cos \\left(3x\\right)-\\cos x}=-\\cot \\left(2x\\right)\\cot x[\/latex]\r\n\r\n58.\u00a0[latex]\\frac{\\cos \\left(6y\\right)+\\cos \\left(8y\\right)}{\\sin \\left(6y\\right)-\\sin \\left(4y\\right)}=\\cot y\\cos \\left(7y\\right)\\sec \\left(5y\\right)[\/latex]\r\n\r\n59. [latex]\\frac{\\cos \\left(2y\\right)-\\cos \\left(4y\\right)}{\\sin \\left(2y\\right)+\\sin \\left(4y\\right)}=\\tan y[\/latex]\r\n\r\n60.\u00a0[latex]\\frac{\\sin \\left(10x\\right)-\\sin \\left(2x\\right)}{\\cos \\left(10x\\right)+\\cos \\left(2x\\right)}=\\tan \\left(4x\\right)[\/latex]\r\n\r\n61. [latex]\\cos x-\\cos \\left(3x\\right)=4{\\sin }^{2}x\\cos x[\/latex]\r\n\r\n62.\u00a0[latex]{\\left(\\cos \\left(2x\\right)-\\cos \\left(4x\\right)\\right)}^{2}+{\\left(\\sin \\left(4x\\right)+\\sin \\left(2x\\right)\\right)}^{2}=4{\\sin }^{2}\\left(3x\\right)[\/latex]\r\n\r\n63. [latex]\\tan \\left(\\frac{\\pi }{4}-t\\right)=\\frac{1-\\tan t}{1+\\tan t}[\/latex]","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400\">Express products as sums.<\/li>\n<li style=\"font-weight: 400\">Express sums as products.<\/li>\n<\/ul>\n<\/div>\n<h2>Expressing Products as Sums<\/h2>\n<p>We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the <strong>product-to-sum formulas<\/strong>, which express products of trigonometric functions as sums. Let\u2019s investigate the cosine identity first and then the sine identity.<\/p>\n<h3>Expressing Products as Sums for Cosine<\/h3>\n<p>We can derive the product-to-sum formula from the sum and difference identities for <strong>cosine<\/strong>. If we add the two equations, we get:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{gathered}\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta =\\cos \\left(\\alpha -\\beta \\right)\\\\\\underline{ +\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta =\\cos \\left(\\alpha +\\beta \\right)} \\\\ 2\\cos \\alpha \\cos \\beta =\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\end{gathered}[\/latex]<\/p>\n<div>Then, we divide by [latex]2[\/latex] to isolate the product of cosines:<\/div>\n<div style=\"text-align: center\">[latex]\\cos \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a product of cosines, express as a sum.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Write the formula for the product of cosines.<\/li>\n<li>Substitute the given angles into the formula.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Writing the Product as a Sum Using the Product-to-Sum Formula for Cosine<\/h3>\n<p>Write the following product of cosines as a sum: [latex]2\\cos \\left(\\frac{7x}{2}\\right)\\cos \\frac{3x}{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q440362\">Show Solution<\/span><\/p>\n<div id=\"q440362\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin by writing the formula for the product of cosines:<\/p>\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\n<p>We can then substitute the given angles into the formula and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}2\\cos \\left(\\frac{7x}{2}\\right)\\cos \\left(\\frac{3x}{2}\\right)&=\\left(2\\right)\\left(\\frac{1}{2}\\right)\\left[\\cos \\left(\\frac{7x}{2}-\\frac{3x}{2}\\right)+\\cos \\left(\\frac{7x}{2}+\\frac{3x}{2}\\right)\\right] \\\\ &=\\left[\\cos \\left(\\frac{4x}{2}\\right)+\\cos \\left(\\frac{10x}{2}\\right)\\right] \\\\ &=\\cos 2x+\\cos 5x \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Use the product-to-sum formula to write the product as a sum or difference: [latex]\\cos \\left(2\\theta \\right)\\cos \\left(4\\theta \\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q293192\">Show Solution<\/span><\/p>\n<div id=\"q293192\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{2}\\left(\\cos 6\\theta +\\cos 2\\theta \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm4642\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4642&theme=oea&iframe_resize_id=ohm4642\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Expressing the Product of Sine and Cosine as a Sum<\/h2>\n<p>Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for <strong>sine<\/strong>. If we add the sum and difference identities, we get:<\/p>\n<div style=\"text-align: center\">[latex]\\begin{gathered}\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta \\\\\\underline{ +\\text{ }\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta}\\\\ \\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)=2\\sin \\alpha \\cos \\beta \\end{gathered}[\/latex]<\/div>\n<p>Then, we divide by 2 to isolate the product of cosine and sine:<\/p>\n<div style=\"text-align: center\">[latex]\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Writing the Product as a Sum Containing only Sine or Cosine<\/h3>\n<p>Express the following product as a sum containing only sine or cosine and no products: [latex]\\sin \\left(4\\theta \\right)\\cos \\left(2\\theta \\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q816809\">Show Solution<\/span><\/p>\n<div id=\"q816809\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\\sin \\alpha \\cos \\beta &=\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right] \\\\ \\sin \\left(4\\theta \\right)\\cos \\left(2\\theta \\right)&=\\frac{1}{2}\\left[\\sin \\left(4\\theta +2\\theta \\right)+\\sin \\left(4\\theta -2\\theta \\right)\\right] \\\\ &=\\frac{1}{2}\\left[\\sin \\left(6\\theta \\right)+\\sin \\left(2\\theta \\right)\\right] \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Use the product-to-sum formula to write the product as a sum: [latex]\\sin \\left(x+y\\right)\\cos \\left(x-y\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q276662\">Show Solution<\/span><\/p>\n<div id=\"q276662\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{2}\\left(\\sin 2x+\\sin 2y\\right)\\\\[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm4631\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4631&theme=oea&iframe_resize_id=ohm4631\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Expressing Products of Sines in Terms of Cosine<\/h2>\n<p>Expressing the product of sines in terms of <strong>cosine<\/strong> is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{gathered}\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta \\\\ \\underline{ -\\text{ }\\cos \\left(\\alpha +\\beta \\right)=-\\left(\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta \\right)} \\\\ \\cos \\left(\\alpha -\\beta \\right)-\\cos \\left(\\alpha +\\beta \\right)=2\\sin \\alpha \\sin \\beta \\end{gathered}[\/latex]<\/p>\n<p>Then, we divide by 2 to isolate the product of sines:<\/p>\n<div style=\"text-align: center\">[latex]\\sin \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)-\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/div>\n<p>Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Product-to-Sum Formulas<\/h3>\n<p>The <strong>product-to-sum formulas<\/strong> are as follows:<\/p>\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\sin \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)-\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)-\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Express the Product as a Sum or Difference<\/h3>\n<p>Write [latex]\\cos \\left(3\\theta \\right)\\cos \\left(5\\theta \\right)[\/latex] as a sum or difference.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q145283\">Show Solution<\/span><\/p>\n<div id=\"q145283\" class=\"hidden-answer\" style=\"display: none\">\n<p>We have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\\cos \\alpha \\cos \\beta &=\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right] \\\\ \\cos \\left(3\\theta \\right)\\cos \\left(5\\theta \\right)&=\\frac{1}{2}\\left[\\cos \\left(3\\theta -5\\theta \\right)+\\cos \\left(3\\theta +5\\theta \\right)\\right] \\\\ &=\\frac{1}{2}\\left[\\cos \\left(2\\theta \\right)+\\cos \\left(8\\theta \\right)\\right] && \\text{Use even-odd identity}. \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Use the product-to-sum formula to evaluate [latex]\\cos \\frac{11\\pi }{12}\\cos \\frac{\\pi }{12}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q731318\">Show Solution<\/span><\/p>\n<div id=\"q731318\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{-2-\\sqrt{3}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Expressing Sums as Products<\/h2>\n<p>Some problems require the reverse of the process we just used. The <strong>sum-to-product formulas<\/strong> allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for <strong>sine<\/strong>. Let [latex]\\frac{u+v}{2}=\\alpha[\/latex] and [latex]\\frac{u-v}{2}=\\beta[\/latex].<\/p>\n<p>Then,<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align}\\alpha +\\beta &=\\frac{u+v}{2}+\\frac{u-v}{2} \\\\ &=\\frac{2u}{2} \\\\ &=u \\\\ \\text{ } \\\\ \\alpha -\\beta &=\\frac{u+v}{2}-\\frac{u-v}{2} \\\\ &=\\frac{2v}{2} \\\\ &=v \\end{align}[\/latex]<\/div>\n<p>Thus, replacing [latex]\\alpha[\/latex] and [latex]\\beta[\/latex] in the product-to-sum formula with the substitute expressions, we have<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align}&\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right] \\\\ &\\sin \\left(\\frac{u+v}{2}\\right)\\cos \\left(\\frac{u-v}{2}\\right)=\\frac{1}{2}\\left[\\sin u+\\sin v\\right]&& \\text{Substitute for}\\left(\\alpha +\\beta \\right)\\text{ and }\\left(\\alpha -\\beta \\right) \\\\ &2\\sin \\left(\\frac{u+v}{2}\\right)\\cos \\left(\\frac{u-v}{2}\\right)=\\sin u+\\sin v \\end{align}[\/latex]<\/div>\n<p>The other sum-to-product identities are derived similarly.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Sum-to-Product Formulas<\/h3>\n<p>The <strong>sum-to-product formulas<\/strong> are as follows:<\/p>\n<p style=\"text-align: center\">[latex]\\sin \\alpha +\\sin \\beta =2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\sin \\alpha -\\sin \\beta =2\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\cos \\alpha -\\cos \\beta =-2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\cos \\alpha +\\cos \\beta =2\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Writing the Difference of Sines as a Product<\/h3>\n<p>Write the following difference of sines expression as a product: [latex]\\sin \\left(4\\theta \\right)-\\sin \\left(2\\theta \\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q627723\">Show Solution<\/span><\/p>\n<div id=\"q627723\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin by writing the formula for the difference of sines.<\/p>\n<p style=\"text-align: center\">[latex]\\sin \\alpha -\\sin \\beta =2\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)[\/latex]<\/p>\n<p>Substitute the values into the formula, and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\\sin \\left(4\\theta \\right)-\\sin \\left(2\\theta \\right)&=2\\sin \\left(\\frac{4\\theta -2\\theta }{2}\\right)\\cos \\left(\\frac{4\\theta +2\\theta }{2}\\right) \\\\ &=2\\sin \\left(\\frac{2\\theta }{2}\\right)\\cos \\left(\\frac{6\\theta }{2}\\right) \\\\ &=2\\sin \\theta \\cos \\left(3\\theta \\right) \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Use the sum-to-product formula to write the sum as a product: [latex]\\sin \\left(3\\theta \\right)+\\sin \\left(\\theta \\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q34084\">Show Solution<\/span><\/p>\n<div id=\"q34084\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2\\sin \\left(2\\theta \\right)\\cos \\left(\\theta \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm4632\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4632&theme=oea&iframe_resize_id=ohm4632\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 5: Evaluating Using the Sum-to-Product Formula<\/h3>\n<p>Evaluate [latex]\\cos \\left({15}^{\\circ }\\right)-\\cos \\left({75}^{\\circ }\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q296492\">Show Solution<\/span><\/p>\n<div id=\"q296492\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin by writing the formula for the difference of cosines.<\/p>\n<p style=\"text-align: center\">[latex]\\cos \\alpha -\\cos \\beta =-2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\n<p>Then we substitute the given angles and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\\cos \\left({15}^{\\circ }\\right)-\\cos \\left({75}^{\\circ }\\right)&=-2\\sin \\left(\\frac{{15}^{\\circ }+{75}^{\\circ }}{2}\\right)\\sin \\left(\\frac{{15}^{\\circ }-{75}^{\\circ }}{2}\\right) \\\\ &=-2\\sin \\left({45}^{\\circ }\\right)\\sin \\left(-{30}^{\\circ }\\right) \\\\ &=-2\\left(\\frac{\\sqrt{2}}{2}\\right)\\left(-\\frac{1}{2}\\right) \\\\ &=\\frac{\\sqrt{2}}{2}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 6: Proving an Identity<\/h3>\n<p>Prove the identity:<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{\\cos \\left(4t\\right)-\\cos \\left(2t\\right)}{\\sin \\left(4t\\right)+\\sin \\left(2t\\right)}=-\\tan t[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q514870\">Show Solution<\/span><\/p>\n<div id=\"q514870\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\\frac{\\cos \\left(4t\\right)-\\cos \\left(2t\\right)}{\\sin \\left(4t\\right)+\\sin \\left(2t\\right)}&=\\frac{-2\\sin \\left(\\frac{4t+2t}{2}\\right)\\sin \\left(\\frac{4t - 2t}{2}\\right)}{2\\sin \\left(\\frac{4t+2t}{2}\\right)\\cos \\left(\\frac{4t - 2t}{2}\\right)} \\\\ &=\\frac{-2\\sin \\left(3t\\right)\\sin t}{2\\sin \\left(3t\\right)\\cos t} \\\\ &=\\frac{-2\\sin \\left(3t\\right)\\sin t}{2\\sin \\left(3t\\right)\\cos t} \\\\ &=-\\frac{\\sin t}{\\cos t} \\\\ &=-\\tan t \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Recall that verifying trigonometric identities has its own set of rules. The procedures for solving an equation are not the same as the procedures for verifying an identity. When we prove an identity, we pick one side to work on and make substitutions until that side is transformed into the other side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities<\/h3>\n<p>Verify the identity [latex]{\\csc }^{2}\\theta -2=\\dfrac{\\cos \\left(2\\theta \\right)}{{\\sin }^{2}\\theta }[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q225762\">Show Solution<\/span><\/p>\n<div id=\"q225762\" class=\"hidden-answer\" style=\"display: none\">\n<p>For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\\frac{\\cos \\left(2\\theta \\right)}{{\\sin }^{2}\\theta }&=\\frac{1 - 2{\\sin }^{2}\\theta }{{\\sin }^{2}\\theta } \\\\ &=\\frac{1}{{\\sin }^{2}\\theta }-\\frac{2{\\sin }^{2}\\theta }{{\\sin }^{2}\\theta } \\\\ &={\\csc }^{2}\\theta -2\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Verify the identity [latex]\\tan \\theta \\cot \\theta -{\\cos }^{2}\\theta ={\\sin }^{2}\\theta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q591097\">Show Solution<\/span><\/p>\n<div id=\"q591097\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align}\\tan \\theta \\cot \\theta -{\\cos }^{2}\\theta &=\\left(\\frac{\\sin \\theta }{\\cos \\theta }\\right)\\left(\\frac{\\cos \\theta }{\\sin \\theta }\\right)-{\\cos }^{2}\\theta \\\\ &=1-{\\cos }^{2}\\theta \\\\ &={\\sin }^{2}\\theta \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<table id=\"fs-id1837805\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/> <\/colgroup>\n<tbody>\n<tr>\n<td><strong>Product-to-sum Formulas<\/strong><\/td>\n<td>\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\sin \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\cos \\left(\\alpha -\\beta \\right)-\\cos \\left(\\alpha +\\beta \\right)\\right][\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\cos \\alpha \\sin \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)-\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td><strong>Sum-to-product Formulas<\/strong><\/td>\n<td>\n<p style=\"text-align: center\">[latex]\\sin \\alpha +\\sin \\beta =2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\sin \\alpha -\\sin \\beta =2\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\cos \\alpha -\\cos \\beta =-2\\sin \\left(\\frac{\\alpha +\\beta }{2}\\right)\\sin \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\cos \\alpha +\\cos \\beta =2\\cos \\left(\\frac{\\alpha +\\beta }{2}\\right)\\cos \\left(\\frac{\\alpha -\\beta }{2}\\right)[\/latex]<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.<\/li>\n<li>We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines.<\/li>\n<li>We can also derive the sum-to-product identities from the product-to-sum identities using substitution.<\/li>\n<li>We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines.<\/li>\n<li>Trigonometric expressions are often simpler to evaluate using the formulas.<\/li>\n<li>The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id2872047\" class=\"definition\">\n<dt>product-to-sum formula<\/dt>\n<dd id=\"fs-id2872050\">a trigonometric identity that allows the writing of a product of trigonometric functions as a sum or difference of trigonometric functions<\/dd>\n<\/dl>\n<dl id=\"fs-id1548089\" class=\"definition\">\n<dt>sum-to-product formula<\/dt>\n<dd id=\"fs-id1548092\">a trigonometric identity that allows, by using substitution, the writing of a sum of trigonometric functions as a product of trigonometric functions<\/dd>\n<\/dl>\n<\/div>\n<p>&nbsp;<\/p>\n<h2 style=\"text-align: center\">Section 5.4 Homework Exercises<\/h2>\n<p>1. Starting with the product to sum formula [latex]\\sin \\alpha \\cos \\beta =\\frac{1}{2}\\left[\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)\\right][\/latex], explain how to determine the formula for [latex]\\cos \\alpha \\sin \\beta[\/latex].<\/p>\n<p>2. Explain two different methods of calculating [latex]\\cos \\left(195^\\circ \\right)\\cos \\left(105^\\circ \\right)[\/latex], one of which uses the product to sum. Which method is easier?<\/p>\n<p>3. Explain a situation where we would convert an equation from a sum to a product and give an example.<\/p>\n<p>4.\u00a0Explain a situation where we would convert an equation from a product to a sum, and give an example.<\/p>\n<p>For the following exercises, rewrite the product as a sum or difference.<\/p>\n<p>5. [latex]16\\sin \\left(16x\\right)\\sin \\left(11x\\right)[\/latex]<\/p>\n<p>6.\u00a0[latex]20\\cos \\left(36t\\right)\\cos \\left(6t\\right)[\/latex]<\/p>\n<p>7. [latex]2\\sin \\left(5x\\right)\\cos \\left(3x\\right)[\/latex]<\/p>\n<p>8.\u00a0[latex]10\\cos \\left(5x\\right)\\sin \\left(10x\\right)[\/latex]<\/p>\n<p>9. [latex]\\sin \\left(-x\\right)\\sin \\left(5x\\right)[\/latex]<\/p>\n<p>10.\u00a0[latex]\\sin \\left(3x\\right)\\cos \\left(5x\\right)[\/latex]<\/p>\n<p>For the following exercises, rewrite the sum or difference as a product.<\/p>\n<p>11. [latex]\\cos \\left(6t\\right)+\\cos \\left(4t\\right)[\/latex]<\/p>\n<p>12.\u00a0[latex]\\sin \\left(3x\\right)+\\sin \\left(7x\\right)[\/latex]<\/p>\n<p>13. [latex]\\cos \\left(7x\\right)+\\cos \\left(-7x\\right)[\/latex]<\/p>\n<p>14.\u00a0[latex]\\sin \\left(3x\\right)-\\sin \\left(-3x\\right)[\/latex]<\/p>\n<p>15. [latex]\\cos \\left(3x\\right)+\\cos \\left(9x\\right)[\/latex]<\/p>\n<p>16.\u00a0[latex]\\sin h-\\sin \\left(3h\\right)[\/latex]<\/p>\n<p>For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.<\/p>\n<p>17. [latex]\\cos \\left(45^\\circ \\right)\\cos \\left(15^\\circ \\right)[\/latex]<\/p>\n<p>18.\u00a0[latex]\\cos \\left(45^\\circ \\right)\\sin \\left(15^\\circ \\right)[\/latex]<\/p>\n<p>19. [latex]\\sin \\left(-345^\\circ \\right)\\sin \\left(-15^\\circ \\right)[\/latex]<\/p>\n<p>20.\u00a0[latex]\\sin \\left(195^\\circ \\right)\\cos \\left(15^\\circ \\right)[\/latex]<\/p>\n<p>21. [latex]\\sin \\left(-45^\\circ \\right)\\sin \\left(-15^\\circ \\right)[\/latex]<\/p>\n<p>For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.<\/p>\n<p>22. [latex]\\cos \\left(23^\\circ \\right)\\sin \\left(17^\\circ \\right)[\/latex]<\/p>\n<p>23. [latex]2\\sin \\left(100^\\circ \\right)\\sin \\left(20^\\circ \\right)[\/latex]<\/p>\n<p>24.\u00a0[latex]2\\sin \\left(-100^\\circ \\right)\\sin \\left(-20^\\circ \\right)[\/latex]<\/p>\n<p>25. [latex]\\sin \\left(213^\\circ \\right)\\cos \\left(8^\\circ \\right)[\/latex]<\/p>\n<p>26.\u00a0[latex]2\\cos \\left(56^\\circ \\right)\\cos \\left(47^\\circ \\right)[\/latex]<\/p>\n<p>For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.<\/p>\n<p>27. [latex]\\sin \\left(76^\\circ \\right)+\\sin \\left(14^\\circ \\right)[\/latex]<\/p>\n<p>28.\u00a0[latex]\\cos \\left(58^\\circ \\right)-\\cos \\left(12^\\circ \\right)[\/latex]<\/p>\n<p>29. [latex]\\sin \\left(101^\\circ \\right)-\\sin \\left(32^\\circ \\right)[\/latex]<\/p>\n<p>30.\u00a0[latex]\\cos \\left(100^\\circ \\right)+\\cos \\left(200^\\circ \\right)[\/latex]<\/p>\n<p>31. [latex]\\sin \\left(-1^\\circ \\right)+\\sin \\left(-2^\\circ \\right)[\/latex]<\/p>\n<p>For the following exercises, prove the identity.<\/p>\n<p>32. [latex]\\frac{\\cos \\left(a+b\\right)}{\\cos \\left(a-b\\right)}=\\frac{1-\\tan a\\tan b}{1+\\tan a\\tan b}[\/latex]<\/p>\n<p>33. [latex]4\\sin \\left(3x\\right)\\cos \\left(4x\\right)=2\\sin \\left(7x\\right)-2\\sin x[\/latex]<\/p>\n<p>34.\u00a0[latex]\\frac{6\\cos \\left(8x\\right)\\sin \\left(2x\\right)}{\\sin \\left(-6x\\right)}=-3\\sin \\left(10x\\right)\\csc \\left(6x\\right)+3[\/latex]<\/p>\n<p>35. [latex]\\sin x+\\sin \\left(3x\\right)=4\\sin x{\\cos }^{2}x[\/latex]<\/p>\n<p>36.\u00a0[latex]2\\left({\\cos }^{3}x-\\cos x{\\sin }^{2}x\\right)=\\cos \\left(3x\\right)+\\cos x[\/latex]<\/p>\n<p>37. [latex]2\\tan x\\cos \\left(3x\\right)=\\sec x\\left(\\sin \\left(4x\\right)-\\sin \\left(2x\\right)\\right)[\/latex]<\/p>\n<p>38. [latex]\\cos \\left(a+b\\right)+\\cos \\left(a-b\\right)=2\\cos a\\cos b[\/latex]<\/p>\n<p>For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.<\/p>\n<p>39. [latex]\\cos \\left({58}^{\\circ }\\right)+\\cos \\left({12}^{\\circ }\\right)[\/latex]<\/p>\n<p>40.\u00a0[latex]\\sin \\left({2}^{\\circ }\\right)-\\sin \\left({3}^{\\circ }\\right)[\/latex]<\/p>\n<p>41. [latex]\\cos \\left({44}^{\\circ }\\right)-\\cos \\left({22}^{\\circ }\\right)[\/latex]<\/p>\n<p>42.\u00a0[latex]\\cos \\left({176}^{\\circ }\\right)\\sin \\left({9}^{\\circ }\\right)[\/latex]<\/p>\n<p>43. [latex]\\sin \\left(-{14}^{\\circ }\\right)\\sin \\left({85}^{\\circ }\\right)[\/latex]<\/p>\n<p>For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.<\/p>\n<p>44. [latex]2\\sin \\left(2x\\right)\\sin \\left(3x\\right)=\\cos x-\\cos \\left(5x\\right)[\/latex]<\/p>\n<p>45. [latex]\\frac{\\cos \\left(10\\theta \\right)+\\cos \\left(6\\theta \\right)}{\\cos \\left(6\\theta \\right)-\\cos \\left(10\\theta \\right)}=\\cot \\left(2\\theta \\right)\\cot \\left(8\\theta \\right)[\/latex]<\/p>\n<p>46.\u00a0[latex]\\frac{\\sin \\left(3x\\right)-\\sin \\left(5x\\right)}{\\cos \\left(3x\\right)+\\cos \\left(5x\\right)}=\\tan x[\/latex]<\/p>\n<p>47. [latex]2\\cos \\left(2x\\right)\\cos x+\\sin \\left(2x\\right)\\sin x=2\\sin x[\/latex]<\/p>\n<p>48.\u00a0[latex]\\frac{\\sin \\left(2x\\right)+\\sin \\left(4x\\right)}{\\sin \\left(2x\\right)-\\sin \\left(4x\\right)}=-\\tan \\left(3x\\right)\\cot x[\/latex]<\/p>\n<p>For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.<\/p>\n<p>49. [latex]\\frac{\\sin \\left(9t\\right)-\\sin \\left(3t\\right)}{\\cos \\left(9t\\right)+\\cos \\left(3t\\right)}[\/latex]<\/p>\n<p>50.\u00a0[latex]2\\sin \\left(8x\\right)\\cos \\left(6x\\right)-\\sin \\left(2x\\right)[\/latex]<\/p>\n<p>51. [latex]\\frac{\\sin \\left(3x\\right)-\\sin x}{\\sin x}[\/latex]<\/p>\n<p>52.\u00a0[latex]\\frac{\\cos \\left(5x\\right)+\\cos \\left(3x\\right)}{\\sin \\left(5x\\right)+\\sin \\left(3x\\right)}[\/latex]<\/p>\n<p>53. [latex]\\sin x\\cos \\left(15x\\right)-\\cos x\\sin \\left(15x\\right)[\/latex]<\/p>\n<p>For the following exercises, prove the following sum-to-product formulas.<\/p>\n<p>54. [latex]\\sin x-\\sin y=2\\sin \\left(\\frac{x-y}{2}\\right)\\cos \\left(\\frac{x+y}{2}\\right)[\/latex]<\/p>\n<p>55. [latex]\\cos x+\\cos y=2\\cos \\left(\\frac{x+y}{2}\\right)\\cos \\left(\\frac{x-y}{2}\\right)[\/latex]<\/p>\n<p>For the following exercises, prove the identity.<\/p>\n<p>56. [latex]\\frac{\\sin \\left(6x\\right)+\\sin \\left(4x\\right)}{\\sin \\left(6x\\right)-\\sin \\left(4x\\right)}=\\tan \\left(5x\\right)\\cot x[\/latex]<\/p>\n<p>57. [latex]\\frac{\\cos \\left(3x\\right)+\\cos x}{\\cos \\left(3x\\right)-\\cos x}=-\\cot \\left(2x\\right)\\cot x[\/latex]<\/p>\n<p>58.\u00a0[latex]\\frac{\\cos \\left(6y\\right)+\\cos \\left(8y\\right)}{\\sin \\left(6y\\right)-\\sin \\left(4y\\right)}=\\cot y\\cos \\left(7y\\right)\\sec \\left(5y\\right)[\/latex]<\/p>\n<p>59. [latex]\\frac{\\cos \\left(2y\\right)-\\cos \\left(4y\\right)}{\\sin \\left(2y\\right)+\\sin \\left(4y\\right)}=\\tan y[\/latex]<\/p>\n<p>60.\u00a0[latex]\\frac{\\sin \\left(10x\\right)-\\sin \\left(2x\\right)}{\\cos \\left(10x\\right)+\\cos \\left(2x\\right)}=\\tan \\left(4x\\right)[\/latex]<\/p>\n<p>61. [latex]\\cos x-\\cos \\left(3x\\right)=4{\\sin }^{2}x\\cos x[\/latex]<\/p>\n<p>62.\u00a0[latex]{\\left(\\cos \\left(2x\\right)-\\cos \\left(4x\\right)\\right)}^{2}+{\\left(\\sin \\left(4x\\right)+\\sin \\left(2x\\right)\\right)}^{2}=4{\\sin }^{2}\\left(3x\\right)[\/latex]<\/p>\n<p>63. [latex]\\tan \\left(\\frac{\\pi }{4}-t\\right)=\\frac{1-\\tan t}{1+\\tan t}[\/latex]<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-14216\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-14216","chapter","type-chapter","status-publish","hentry"],"part":14191,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/14216","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/14216\/revisions"}],"predecessor-version":[{"id":17605,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/14216\/revisions\/17605"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/parts\/14191"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/14216\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/media?parent=14216"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=14216"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/contributor?post=14216"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/license?post=14216"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}