{"id":1980,"date":"2021-08-19T16:03:44","date_gmt":"2021-08-19T16:03:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-calculus-of-the-hyperbolic-functions\/"},"modified":"2021-11-17T01:53:51","modified_gmt":"2021-11-17T01:53:51","slug":"skills-review-for-calculus-of-the-hyperbolic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-calculus-of-the-hyperbolic-functions\/","title":{"raw":"Skills Review for Calculus of the Hyperbolic Functions","rendered":"Skills Review for Calculus of the Hyperbolic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Verify the fundamental trigonometric identities.&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:14913,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Verify the fundamental trigonometric identities<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Calculus of the Hyperbolic Functions section, we will learn how to differentiate and integrate hyperbolic functions. Here we will review some trigonometric formulas and how to use them. Some of these formulas are identical to those used for hyperbolic functions.\r\n<h2>Apply Trigonometric Formulas<\/h2>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sum and Difference Formulas for Cosine<\/h3>\r\nThe <strong>sum and difference formulas for cosine<\/strong> are:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha +\\beta \\right)=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta\\end{align} [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta\\end{align} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sum and Difference Formulas for Sine<\/h3>\r\nThe <strong>sum and difference formulas for sine<\/strong> are:\r\n<p style=\"text-align: center;\">[latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta [\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sum and Difference Formulas for Tangent<\/h3>\r\nThe <strong>sum and difference formulas for tangent<\/strong> are:\r\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\alpha +\\beta \\right)=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta }[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\alpha -\\beta \\right)=\\frac{\\tan \\alpha -\\tan \\beta }{1+\\tan \\alpha \\tan \\beta }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an identity, verify using sum and difference formulas<\/h3>\r\n<ol>\r\n \t<li>Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.<\/li>\r\n \t<li>Look for opportunities to use the sum and difference formulas.<\/li>\r\n \t<li>Rewrite sums or differences of quotients as single quotients.<\/li>\r\n \t<li>If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying Trigonometric Formulas to Verify Trigonometric Identities<\/h3>\r\nVerify the identity [latex]\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)=2\\sin \\alpha \\cos \\beta [\/latex].\r\n\r\n[reveal-answer q=\"464113\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"464113\"]\r\n\r\nWe see that the left side of the equation includes the sines of the sum and the difference of angles.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta\\\\ \\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\end{gathered}[\/latex]<\/p>\r\nWe can rewrite each using the sum and difference formulas.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)&amp;=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta +\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\\\ &amp;=2\\sin \\alpha \\cos \\beta \\end{align}[\/latex]<\/p>\r\nWe see that the identity is verified.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying Trigonometric Formulas to Verify Trigonometric Identities<\/h3>\r\nVerify the following identity.\r\n<p style=\"text-align: center;\">[latex]\\frac{\\sin \\left(\\alpha -\\beta \\right)}{\\cos \\alpha \\cos \\beta }=\\tan \\alpha -\\tan \\beta [\/latex]<\/p>\r\n[reveal-answer q=\"605610\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"605610\"]\r\n\r\nWe can begin by rewriting the numerator on the left side of the equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\sin \\left(\\alpha -\\beta \\right)}{\\cos \\alpha \\cos \\beta }&amp;=\\frac{\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta } \\\\ &amp;=\\frac{\\sin \\alpha \\cos \\beta}{\\cos \\alpha \\cos \\beta}-\\frac{\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta } &amp;&amp; \\text{Rewrite using a common denominator}. \\\\ &amp;=\\frac{\\sin \\alpha }{\\cos \\alpha }-\\frac{\\sin \\beta }{\\cos \\beta }&amp;&amp; \\text{Cancel}. \\\\ &amp;=\\tan \\alpha -\\tan \\beta &amp;&amp; \\text{Rewrite in terms of tangent}.\\end{align}[\/latex]<\/p>\r\nWe see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nVerify the identity: [latex]\\tan \\left(\\pi -\\theta \\right)=-\\tan \\theta [\/latex].\r\n\r\n[reveal-answer q=\"184004\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"184004\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\pi -\\theta \\right)&amp;=\\frac{\\tan \\left(\\pi \\right)-\\tan \\theta }{1+\\tan \\left(\\pi \\right)\\tan \\theta } \\\\ &amp;=\\frac{0-\\tan \\theta }{1+0\\cdot \\tan \\theta } \\\\ &amp;=-\\tan \\theta \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Double-Angle Formulas<\/h3>\r\nThe <strong>double-angle formulas<\/strong> are summarized as follows:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(2\\theta \\right)&amp;=2\\sin \\theta \\cos \\theta\\\\\\text{ }\\\\ \\cos \\left(2\\theta \\right)&amp;={\\cos }^{2}\\theta -{\\sin }^{2}\\theta \\\\ &amp;=1 - 2{\\sin }^{2}\\theta \\\\ &amp;=2{\\cos }^{2}\\theta -1 \\\\\\text{ }\\\\ \\tan \\left(2\\theta \\right)&amp;=\\frac{2\\tan \\theta }{1-{\\tan }^{2}\\theta }\\end{align}[\/latex]<\/p>\r\n\r\n<\/div>\r\nEstablishing identities using the double-angle formulas is performed using the same steps we used to establish identities using the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying Trigonometric Formulas to Verify Trigonometric Identities<\/h3>\r\nEstablish the following identity using double-angle formulas:\r\n<p style=\"text-align: center;\">[latex]1+\\sin \\left(2\\theta \\right)={\\left(\\sin \\theta +\\cos \\theta \\right)}^{2}[\/latex]<\/p>\r\n[reveal-answer q=\"600157\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"600157\"]\r\n\r\nWe will work on the right side of the equal sign and rewrite the expression until it matches the left side.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{\\left(\\sin \\theta +\\cos \\theta \\right)}^{2}&amp;={\\sin }^{2}\\theta +2\\sin \\theta \\cos \\theta +{\\cos }^{2}\\theta \\\\ &amp;=\\left({\\sin }^{2}\\theta +{\\cos }^{2}\\theta \\right)+2\\sin \\theta \\cos \\theta \\\\ &amp;=1+2\\sin \\theta \\cos \\theta \\\\ &amp;=1+\\sin \\left(2\\theta \\right)\\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThis process is not complicated, as long as we recall the perfect square formula from algebra:\r\n<p style=\"text-align: center;\">[latex]{\\left(a\\pm b\\right)}^{2}={a}^{2}\\pm 2ab+{b}^{2}[\/latex]<\/p>\r\nwhere [latex]a=\\sin \\theta [\/latex] and [latex]b=\\cos \\theta [\/latex]. Part of being successful in mathematics is the ability to recognize patterns. While the terms or symbols may change, the algebra remains consistent.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nEstablish the identity: [latex]{\\cos }^{4}\\theta -{\\sin }^{4}\\theta =\\cos \\left(2\\theta \\right)[\/latex].\r\n\r\n[reveal-answer q=\"178499\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"178499\"]\r\n\r\n[latex]{\\cos }^{4}\\theta -{\\sin }^{4}\\theta =\\left({\\cos }^{2}\\theta +{\\sin }^{2}\\theta \\right)\\left({\\cos }^{2}\\theta -{\\sin }^{2}\\theta \\right)=\\cos \\left(2\\theta \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying Trigonometric Formulas to Verify Trigonometric Identities<\/h3>\r\nVerify the identity:\r\n<p style=\"text-align: center;\">[latex]\\tan \\left(2\\theta \\right)=\\frac{2}{\\cot \\theta -\\tan \\theta }[\/latex]<\/p>\r\n[reveal-answer q=\"395376\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"395376\"]\r\n\r\nIn this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(2\\theta \\right)&amp;=\\frac{2\\tan \\theta }{1-{\\tan }^{2}\\theta }&amp;&amp; \\text{Double-angle formula} \\\\ &amp;=\\frac{2\\tan \\theta \\left(\\frac{1}{\\tan \\theta }\\right)}{\\left(1-{\\tan }^{2}\\theta \\right)\\left(\\frac{1}{\\tan \\theta }\\right)}&amp;&amp; \\text{Multiply by a term that results in desired numerator}. \\\\ &amp;=\\frac{2}{\\frac{1}{\\tan \\theta }-\\frac{{\\tan }^{2}\\theta }{\\tan \\theta }} \\\\ &amp;=\\frac{2}{\\cot \\theta -\\tan \\theta }&amp;&amp; \\text{Use reciprocal identity for }\\frac{1}{\\tan \\theta }.\\end{align}[\/latex]<\/p>\r\n\r\n<h4 style=\"text-align: left;\">Analysis of the Solution<\/h4>\r\nHere is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the left side. However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the equivalency. For example, suppose that we wanted to show\r\n<p style=\"text-align: center;\">[latex]\\frac{2\\tan \\theta }{1-{\\tan }^{2}\\theta }=\\frac{2}{\\cot \\theta -\\tan \\theta }[\/latex]<\/p>\r\nLet\u2019s work on the right side.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{2}{\\cot \\theta -\\tan \\theta }&amp;=\\frac{2}{\\frac{1}{\\tan \\theta }-\\tan \\theta }\\left(\\frac{\\tan \\theta }{\\tan \\theta }\\right) \\\\ &amp;=\\frac{2\\tan \\theta }{\\frac{1}{\\cancel{\\tan \\theta }}\\left(\\cancel{\\tan \\theta }\\right)-\\tan \\theta \\left(\\tan \\theta \\right)} \\\\ &amp;=\\frac{2\\tan \\theta }{1-{\\tan }^{2}\\theta } \\end{align}[\/latex]<\/p>\r\nWhen using the identities to simplify a trigonometric expression or solve a trigonometric equation, there are usually several paths to a desired result. There is no set rule as to what side should be manipulated. However, we should begin with the guidelines set forth earlier.\r\n\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]\\cos \\left(2\\theta \\right)\\cos \\theta ={\\cos }^{3}\\theta -\\cos \\theta {\\sin }^{2}\\theta [\/latex].\r\n\r\n[reveal-answer q=\"584630\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"584630\"]\r\n\r\n[latex]\\cos \\left(2\\theta \\right)\\cos \\theta =\\left({\\cos }^{2}\\theta -{\\sin }^{2}\\theta \\right)\\cos \\theta ={\\cos }^{3}\\theta -\\cos \\theta {\\sin }^{2}\\theta [\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Verify the fundamental trigonometric identities.&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:14913,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Verify the fundamental trigonometric identities<\/span><\/li>\n<\/ul>\n<\/div>\n<p>In the Calculus of the Hyperbolic Functions section, we will learn how to differentiate and integrate hyperbolic functions. Here we will review some trigonometric formulas and how to use them. Some of these formulas are identical to those used for hyperbolic functions.<\/p>\n<h2>Apply Trigonometric Formulas<\/h2>\n<div class=\"textbox\">\n<h3>A General Note: Sum and Difference Formulas for Cosine<\/h3>\n<p>The <strong>sum and difference formulas for cosine<\/strong> are:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha +\\beta \\right)=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta\\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha -\\beta \\right)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta\\end{align}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Sum and Difference Formulas for Sine<\/h3>\n<p>The <strong>sum and difference formulas for sine<\/strong> are:<\/p>\n<p style=\"text-align: center;\">[latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Sum and Difference Formulas for Tangent<\/h3>\n<p>The <strong>sum and difference formulas for tangent<\/strong> are:<\/p>\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\alpha +\\beta \\right)=\\frac{\\tan \\alpha +\\tan \\beta }{1-\\tan \\alpha \\tan \\beta }[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\tan \\left(\\alpha -\\beta \\right)=\\frac{\\tan \\alpha -\\tan \\beta }{1+\\tan \\alpha \\tan \\beta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an identity, verify using sum and difference formulas<\/h3>\n<ol>\n<li>Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.<\/li>\n<li>Look for opportunities to use the sum and difference formulas.<\/li>\n<li>Rewrite sums or differences of quotients as single quotients.<\/li>\n<li>If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying Trigonometric Formulas to Verify Trigonometric Identities<\/h3>\n<p>Verify the identity [latex]\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)=2\\sin \\alpha \\cos \\beta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q464113\">Show Solution<\/span><\/p>\n<div id=\"q464113\" class=\"hidden-answer\" style=\"display: none\">\n<p>We see that the left side of the equation includes the sines of the sum and the difference of angles.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta\\\\ \\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\end{gathered}[\/latex]<\/p>\n<p>We can rewrite each using the sum and difference formulas.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(\\alpha +\\beta \\right)+\\sin \\left(\\alpha -\\beta \\right)&=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta +\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\\\ &=2\\sin \\alpha \\cos \\beta \\end{align}[\/latex]<\/p>\n<p>We see that the identity is verified.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying Trigonometric Formulas to Verify Trigonometric Identities<\/h3>\n<p>Verify the following identity.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\sin \\left(\\alpha -\\beta \\right)}{\\cos \\alpha \\cos \\beta }=\\tan \\alpha -\\tan \\beta[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q605610\">Show Solution<\/span><\/p>\n<div id=\"q605610\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can begin by rewriting the numerator on the left side of the equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\sin \\left(\\alpha -\\beta \\right)}{\\cos \\alpha \\cos \\beta }&=\\frac{\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta } \\\\ &=\\frac{\\sin \\alpha \\cos \\beta}{\\cos \\alpha \\cos \\beta}-\\frac{\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta } && \\text{Rewrite using a common denominator}. \\\\ &=\\frac{\\sin \\alpha }{\\cos \\alpha }-\\frac{\\sin \\beta }{\\cos \\beta }&& \\text{Cancel}. \\\\ &=\\tan \\alpha -\\tan \\beta && \\text{Rewrite in terms of tangent}.\\end{align}[\/latex]<\/p>\n<p>We see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.<\/p>\n<\/div>\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 \\left(\\pi -\\theta \\right)=-\\tan \\theta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q184004\">Show Solution<\/span><\/p>\n<div id=\"q184004\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(\\pi -\\theta \\right)&=\\frac{\\tan \\left(\\pi \\right)-\\tan \\theta }{1+\\tan \\left(\\pi \\right)\\tan \\theta } \\\\ &=\\frac{0-\\tan \\theta }{1+0\\cdot \\tan \\theta } \\\\ &=-\\tan \\theta \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Double-Angle Formulas<\/h3>\n<p>The <strong>double-angle formulas<\/strong> are summarized as follows:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sin \\left(2\\theta \\right)&=2\\sin \\theta \\cos \\theta\\\\\\text{ }\\\\ \\cos \\left(2\\theta \\right)&={\\cos }^{2}\\theta -{\\sin }^{2}\\theta \\\\ &=1 - 2{\\sin }^{2}\\theta \\\\ &=2{\\cos }^{2}\\theta -1 \\\\\\text{ }\\\\ \\tan \\left(2\\theta \\right)&=\\frac{2\\tan \\theta }{1-{\\tan }^{2}\\theta }\\end{align}[\/latex]<\/p>\n<\/div>\n<p>Establishing identities using the double-angle formulas is performed using the same steps we used to establish identities using the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Applying Trigonometric Formulas to Verify Trigonometric Identities<\/h3>\n<p>Establish the following identity using double-angle formulas:<\/p>\n<p style=\"text-align: center;\">[latex]1+\\sin \\left(2\\theta \\right)={\\left(\\sin \\theta +\\cos \\theta \\right)}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q600157\">Show Solution<\/span><\/p>\n<div id=\"q600157\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will work on the right side of the equal sign and rewrite the expression until it matches the left side.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{\\left(\\sin \\theta +\\cos \\theta \\right)}^{2}&={\\sin }^{2}\\theta +2\\sin \\theta \\cos \\theta +{\\cos }^{2}\\theta \\\\ &=\\left({\\sin }^{2}\\theta +{\\cos }^{2}\\theta \\right)+2\\sin \\theta \\cos \\theta \\\\ &=1+2\\sin \\theta \\cos \\theta \\\\ &=1+\\sin \\left(2\\theta \\right)\\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This process is not complicated, as long as we recall the perfect square formula from algebra:<\/p>\n<p style=\"text-align: center;\">[latex]{\\left(a\\pm b\\right)}^{2}={a}^{2}\\pm 2ab+{b}^{2}[\/latex]<\/p>\n<p>where [latex]a=\\sin \\theta[\/latex] and [latex]b=\\cos \\theta[\/latex]. Part of being successful in mathematics is the ability to recognize patterns. While the terms or symbols may change, the algebra remains consistent.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Establish the identity: [latex]{\\cos }^{4}\\theta -{\\sin }^{4}\\theta =\\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=\"q178499\">Show Solution<\/span><\/p>\n<div id=\"q178499\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{\\cos }^{4}\\theta -{\\sin }^{4}\\theta =\\left({\\cos }^{2}\\theta +{\\sin }^{2}\\theta \\right)\\left({\\cos }^{2}\\theta -{\\sin }^{2}\\theta \\right)=\\cos \\left(2\\theta \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying Trigonometric Formulas to Verify Trigonometric Identities<\/h3>\n<p>Verify the identity:<\/p>\n<p style=\"text-align: center;\">[latex]\\tan \\left(2\\theta \\right)=\\frac{2}{\\cot \\theta -\\tan \\theta }[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q395376\">Show Solution<\/span><\/p>\n<div id=\"q395376\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan \\left(2\\theta \\right)&=\\frac{2\\tan \\theta }{1-{\\tan }^{2}\\theta }&& \\text{Double-angle formula} \\\\ &=\\frac{2\\tan \\theta \\left(\\frac{1}{\\tan \\theta }\\right)}{\\left(1-{\\tan }^{2}\\theta \\right)\\left(\\frac{1}{\\tan \\theta }\\right)}&& \\text{Multiply by a term that results in desired numerator}. \\\\ &=\\frac{2}{\\frac{1}{\\tan \\theta }-\\frac{{\\tan }^{2}\\theta }{\\tan \\theta }} \\\\ &=\\frac{2}{\\cot \\theta -\\tan \\theta }&& \\text{Use reciprocal identity for }\\frac{1}{\\tan \\theta }.\\end{align}[\/latex]<\/p>\n<h4 style=\"text-align: left;\">Analysis of the Solution<\/h4>\n<p>Here is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the left side. However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the equivalency. For example, suppose that we wanted to show<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2\\tan \\theta }{1-{\\tan }^{2}\\theta }=\\frac{2}{\\cot \\theta -\\tan \\theta }[\/latex]<\/p>\n<p>Let\u2019s work on the right side.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{2}{\\cot \\theta -\\tan \\theta }&=\\frac{2}{\\frac{1}{\\tan \\theta }-\\tan \\theta }\\left(\\frac{\\tan \\theta }{\\tan \\theta }\\right) \\\\ &=\\frac{2\\tan \\theta }{\\frac{1}{\\cancel{\\tan \\theta }}\\left(\\cancel{\\tan \\theta }\\right)-\\tan \\theta \\left(\\tan \\theta \\right)} \\\\ &=\\frac{2\\tan \\theta }{1-{\\tan }^{2}\\theta } \\end{align}[\/latex]<\/p>\n<p>When using the identities to simplify a trigonometric expression or solve a trigonometric equation, there are usually several paths to a desired result. There is no set rule as to what side should be manipulated. However, we should begin with the guidelines set forth earlier.<\/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]\\cos \\left(2\\theta \\right)\\cos \\theta ={\\cos }^{3}\\theta -\\cos \\theta {\\sin }^{2}\\theta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q584630\">Show Solution<\/span><\/p>\n<div id=\"q584630\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\cos \\left(2\\theta \\right)\\cos \\theta =\\left({\\cos }^{2}\\theta -{\\sin }^{2}\\theta \\right)\\cos \\theta ={\\cos }^{3}\\theta -\\cos \\theta {\\sin }^{2}\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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-1980\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/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":349141,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision \",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen 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