{"id":15699,"date":"2019-09-05T17:58:13","date_gmt":"2019-09-05T17:58:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=15699"},"modified":"2025-02-05T05:23:34","modified_gmt":"2025-02-05T05:23:34","slug":"problem-set-54-sum-to-product-and-product-to-sum-formulas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-54-sum-to-product-and-product-to-sum-formulas\/","title":{"raw":"Problem Set 54: Sum-to-Product and Product-to-Sum Formulas","rendered":"Problem Set 54: Sum-to-Product and Product-to-Sum Formulas"},"content":{"raw":"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].\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":"<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-15699\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <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>. <strong>License Terms<\/strong>: Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/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":169554,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15699","chapter","type-chapter","status-publish","hentry"],"part":14191,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15699","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15699\/revisions"}],"predecessor-version":[{"id":15701,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15699\/revisions\/15701"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/14191"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15699\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=15699"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=15699"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=15699"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=15699"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}