{"id":205,"date":"2025-02-13T22:45:01","date_gmt":"2025-02-13T22:45:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sum-to-product-and-product-to-sum-formulas\/"},"modified":"2025-10-16T17:28:17","modified_gmt":"2025-10-16T17:28:17","slug":"sum-to-product-and-product-to-sum-formulas","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sum-to-product-and-product-to-sum-formulas\/","title":{"raw":"Sum-to-Product and Product-to-Sum Formulas: Learn It 1","rendered":"Sum-to-Product and Product-to-Sum Formulas: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\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<\/section>\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><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>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<\/section><\/div>\r\n<div><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a product of cosines, express as a sum.\r\n<\/strong>\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<\/section><\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write the following product of cosines as a sum: [latex]2\\cos \\left(\\frac{7x}{2}\\right)\\cos \\frac{3x}{2}[\/latex].[reveal-answer q=\"440362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"440362\"]We 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<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">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].[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][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]4642[\/ohm_question]<\/section>\r\n<h3>Expressing the Product of Sine and Cosine as a Sum<\/h3>\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><section class=\"textbox example\" aria-label=\"Example\">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].[reveal-answer q=\"816809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"816809\"]Write 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<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">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].[reveal-answer q=\"276662\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"276662\"][latex]\\frac{1}{2}\\left(\\sin 2x+\\sin 2y\\right)\\\\[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]4631[\/ohm_question]<\/section><\/div>\r\n<section aria-label=\"Try It\">\r\n<h3>Expressing Products of Sines in Terms of Cosine<\/h3>\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\r\n<section class=\"textbox example\" aria-label=\"Example\">Write [latex]\\cos \\left(3\\theta \\right)\\cos \\left(5\\theta \\right)[\/latex] as a sum or difference.[reveal-answer q=\"145283\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145283\"]We 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<\/section>\r\n<div class=\"bcc-box bcc-success\"><section class=\"textbox tryIt\" aria-label=\"Try It\">Use the product-to-sum formula to evaluate [latex]\\cos \\frac{11\\pi }{12}\\cos \\frac{\\pi }{12}[\/latex].[reveal-answer q=\"731318\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"731318\"][latex]\\frac{-2-\\sqrt{3}}{4}[\/latex][\/hidden-answer]<\/section><\/div>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\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<\/section>\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>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>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<\/section>\n<\/div>\n<div>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a product of cosines, express as a sum.<br \/>\n<\/strong><\/p>\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<\/section>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q440362\">Show Solution<\/button><\/p>\n<div id=\"q440362\" class=\"hidden-answer\" style=\"display: none\">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<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q293192\">Show Solution<\/button><\/p>\n<div id=\"q293192\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{2}\\left(\\cos 6\\theta +\\cos 2\\theta \\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm4642\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4642&theme=lumen&iframe_resize_id=ohm4642&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Expressing the Product of Sine and Cosine as a Sum<\/h3>\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>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q816809\">Show Solution<\/button><\/p>\n<div id=\"q816809\" class=\"hidden-answer\" style=\"display: none\">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<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q276662\">Show Solution<\/button><\/p>\n<div id=\"q276662\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{2}\\left(\\sin 2x+\\sin 2y\\right)\\\\[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm4631\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4631&theme=lumen&iframe_resize_id=ohm4631&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<section aria-label=\"Try It\">\n<h3>Expressing Products of Sines in Terms of Cosine<\/h3>\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<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q145283\">Show Solution<\/button><\/p>\n<div id=\"q145283\" class=\"hidden-answer\" style=\"display: none\">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<\/section>\n<div class=\"bcc-box bcc-success\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q731318\">Show Solution<\/button><\/p>\n<div id=\"q731318\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{-2-\\sqrt{3}}{4}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":22,"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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":201,"module-header":"learn_it","content_attributions":[{"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":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/205"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/205\/revisions"}],"predecessor-version":[{"id":4685,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/205\/revisions\/4685"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/201"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/205\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=205"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=205"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=205"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=205"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}