{"id":1533,"date":"2025-07-25T02:37:51","date_gmt":"2025-07-25T02:37:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1533"},"modified":"2026-03-12T06:03:27","modified_gmt":"2026-03-12T06:03:27","slug":"sum-and-difference-identities-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sum-and-difference-identities-fresh-take\/","title":{"raw":"Sum and Difference Identities: Fresh Take","rendered":"Sum and Difference Identities: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use sum and difference formulas for sine, cosine, and tangent<\/li>\r\n \t<li>Use sum and difference formulas to verify identities.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Sum and Difference Formulas<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"56\" data-end=\"536\">Sum and difference formulas allow us to find the sine, cosine, or tangent of non-special angles by rewriting them in terms of sums or differences of angles we know. These formulas also let us simplify trig expressions and prove identities. They are especially helpful for evaluating angles like [latex]75^\\circ[\/latex] or [latex]15^\\circ[\/latex], since those can be written as sums or differences of [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], and [latex]60^\\circ[\/latex].<\/p>\r\n<p data-start=\"538\" data-end=\"557\">The formulas are:<\/p>\r\n\r\n<ul data-start=\"559\" data-end=\"774\">\r\n \t<li data-start=\"559\" data-end=\"625\">\r\n<p data-start=\"561\" data-end=\"625\">[latex]\\sin(A \\pm B) = \\sin A \\cos B \\pm \\cos A \\sin B[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"626\" data-end=\"692\">\r\n<p data-start=\"628\" data-end=\"692\">[latex]\\cos(A \\pm B) = \\cos A \\cos B \\mp \\sin A \\sin B[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"693\" data-end=\"774\">\r\n<p data-start=\"695\" data-end=\"774\">[latex]\\tan(A \\pm B) = \\dfrac{\\tan A \\pm \\tan B}{1 \\mp \\tan A \\tan B}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Sum and Difference Formulas<\/strong>\r\n<ol>\r\n \t<li data-start=\"834\" data-end=\"1079\">\r\n<p data-start=\"837\" data-end=\"864\"><strong data-start=\"837\" data-end=\"862\">Remember the Patterns<\/strong><\/p>\r\n\r\n<ul data-start=\"868\" data-end=\"1079\">\r\n \t<li data-start=\"868\" data-end=\"933\">\r\n<p data-start=\"870\" data-end=\"933\">Sine keeps the same sign: plus for sum, minus for difference.<\/p>\r\n<\/li>\r\n \t<li data-start=\"937\" data-end=\"999\">\r\n<p data-start=\"939\" data-end=\"999\">Cosine flips the sign: plus for difference, minus for sum.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1003\" data-end=\"1079\">\r\n<p data-start=\"1005\" data-end=\"1079\">Tangent has the same sign in the numerator, opposite in the denominator.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1081\" data-end=\"1346\">\r\n<p data-start=\"1084\" data-end=\"1108\"><strong data-start=\"1084\" data-end=\"1106\">Use Special Angles<\/strong><\/p>\r\n\r\n<ul data-start=\"1112\" data-end=\"1346\">\r\n \t<li data-start=\"1112\" data-end=\"1275\">\r\n<p data-start=\"1114\" data-end=\"1275\">Break down unfamiliar angles into sums or differences of [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], [latex]60^\\circ[\/latex], or [latex]90^\\circ[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1279\" data-end=\"1346\">\r\n<p data-start=\"1281\" data-end=\"1346\">Example: [latex]\\cos(75^\\circ)=\\cos(45^\\circ+30^\\circ)[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1718\" data-end=\"1869\">\r\n<p data-start=\"1721\" data-end=\"1750\"><strong data-start=\"1721\" data-end=\"1748\">Check Signs by Quadrant<\/strong><\/p>\r\n\r\n<ul data-start=\"1896\" data-end=\"2020\">\r\n \t<li data-start=\"1754\" data-end=\"1869\">\r\n<p data-start=\"1756\" data-end=\"1869\">Always be mindful of whether the angle is in a quadrant where sine, cosine, or tangent is positive or negative.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Find the exact value of [latex]\\sin(75\u00b0)[\/latex] using a sum formula.[reveal-answer q=\"sum-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"sum-001\"]\r\nWrite [latex]75\u00b0 = 45\u00b0 + 30\u00b0[\/latex]Use the sum formula: [latex]\\sin(\\alpha + \\beta) = \\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta[\/latex][latex]\\begin{align*}\r\n\\sin(75\u00b0) &amp;= \\sin(45\u00b0 + 30\u00b0) \\\\\r\n&amp;= \\sin 45\u00b0 \\cos 30\u00b0 + \\cos 45\u00b0 \\sin 30\u00b0 \\\\\r\n&amp;= \\frac{\\sqrt{2}}{2} \\cdot \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{2}}{2} \\cdot \\frac{1}{2} \\\\\r\n&amp;= \\frac{\\sqrt{6}}{4} + \\frac{\\sqrt{2}}{4} \\\\\r\n&amp;= \\frac{\\sqrt{6} + \\sqrt{2}}{4}\r\n\\end{align*}[\/latex]\r\n[\/hidden-answer]<\/section>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-beacchfg-liSgx_72bfg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/liSgx_72bfg?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-beacchfg-liSgx_72bfg\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661365&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-beacchfg-liSgx_72bfg&vembed=0&video_id=liSgx_72bfg&video_target=tpm-plugin-beacchfg-liSgx_72bfg'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Using+sum+and+difference+formula+to+find+the+exact+value+with+cosine_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUsing sum and difference formula to find the exact value with cosine\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Verifying Identities with Sum and Difference Formulas<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"82\" data-end=\"402\">Sum and difference formulas are powerful tools for proving that two trigonometric expressions are equal. When verifying identities, you often expand one side using these formulas, then simplify until it matches the other side. This process shows that the two sides are equivalent for all values where both are defined.<\/p>\r\n<p data-start=\"404\" data-end=\"423\">The formulas are:<\/p>\r\n\r\n<ul data-start=\"425\" data-end=\"640\">\r\n \t<li data-start=\"425\" data-end=\"491\">\r\n<p data-start=\"427\" data-end=\"491\">[latex]\\sin(A \\pm B) = \\sin A \\cos B \\pm \\cos A \\sin B[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"492\" data-end=\"558\">\r\n<p data-start=\"494\" data-end=\"558\">[latex]\\cos(A \\pm B) = \\cos A \\cos B \\mp \\sin A \\sin B[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"559\" data-end=\"640\">\r\n<p data-start=\"561\" data-end=\"640\">[latex]\\tan(A \\pm B) = \\dfrac{\\tan A \\pm \\tan B}{1 \\mp \\tan A \\tan B}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Sum and Difference Formulas in Verifications<\/strong>\r\n<ol>\r\n \t<li data-start=\"717\" data-end=\"865\">\r\n<p data-start=\"720\" data-end=\"748\"><strong data-start=\"720\" data-end=\"746\">Expand Using a Formula<\/strong><\/p>\r\n\r\n<ul data-start=\"752\" data-end=\"865\">\r\n \t<li data-start=\"752\" data-end=\"809\">\r\n<p data-start=\"754\" data-end=\"809\">Start with the more complicated side of the identity.<\/p>\r\n<\/li>\r\n \t<li data-start=\"813\" data-end=\"865\">\r\n<p data-start=\"815\" data-end=\"865\">Apply the appropriate sum or difference formula.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"867\" data-end=\"1037\">\r\n<p data-start=\"870\" data-end=\"894\"><strong data-start=\"870\" data-end=\"892\">Simplify Carefully<\/strong><\/p>\r\n\r\n<ul data-start=\"898\" data-end=\"1037\">\r\n \t<li data-start=\"898\" data-end=\"971\">\r\n<p data-start=\"900\" data-end=\"971\">Replace sine, cosine, or tangent of special angles with exact values.<\/p>\r\n<\/li>\r\n \t<li data-start=\"975\" data-end=\"1037\">\r\n<p data-start=\"977\" data-end=\"1037\">Combine like terms, simplify fractions, or cancel factors.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<ul data-start=\"1172\" data-end=\"1334\">\r\n \t<li data-start=\"1304\" data-end=\"1334\">\r\n<p data-start=\"1339\" data-end=\"1372\"><strong data-start=\"1339\" data-end=\"1370\">Look for Target Expressions<\/strong><\/p>\r\n\r\n<ul data-start=\"1376\" data-end=\"1510\">\r\n \t<li data-start=\"1376\" data-end=\"1510\">\r\n<p data-start=\"1378\" data-end=\"1510\">If the identity involves a single trig function of a sum or difference, try collapsing products into one cosine, sine, or tangent.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Verify the identity: [latex]\\sin(x + \\pi) = -\\sin x[\/latex][reveal-answer q=\"sum-002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"sum-002\"]\r\nUse the sum formula: [latex]\\sin(\\alpha + \\beta) = \\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta[\/latex][latex]\\begin{align*}\r\n\\sin(x + \\pi) &amp;= \\sin x \\cos \\pi + \\cos x \\sin \\pi \\\\\r\n&amp;= \\sin x (-1) + \\cos x (0) \\\\\r\n&amp;= -\\sin x + 0 \\\\\r\n&amp;= -\\sin x\r\n\\end{align*}[\/latex]The identity is verified.\r\n[\/hidden-answer]<\/section>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-befeagae-qMcYCXaveRM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/qMcYCXaveRM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-befeagae-qMcYCXaveRM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661366&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-befeagae-qMcYCXaveRM&vembed=0&video_id=qMcYCXaveRM&video_target=tpm-plugin-befeagae-qMcYCXaveRM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+verify+an+identity+with+the+sum+and+difference+formulas_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to verify an identity with the sum and difference formulas\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use sum and difference formulas for sine, cosine, and tangent<\/li>\n<li>Use sum and difference formulas to verify identities.<\/li>\n<\/ul>\n<\/section>\n<h2>Sum and Difference Formulas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"56\" data-end=\"536\">Sum and difference formulas allow us to find the sine, cosine, or tangent of non-special angles by rewriting them in terms of sums or differences of angles we know. These formulas also let us simplify trig expressions and prove identities. They are especially helpful for evaluating angles like [latex]75^\\circ[\/latex] or [latex]15^\\circ[\/latex], since those can be written as sums or differences of [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], and [latex]60^\\circ[\/latex].<\/p>\n<p data-start=\"538\" data-end=\"557\">The formulas are:<\/p>\n<ul data-start=\"559\" data-end=\"774\">\n<li data-start=\"559\" data-end=\"625\">\n<p data-start=\"561\" data-end=\"625\">[latex]\\sin(A \\pm B) = \\sin A \\cos B \\pm \\cos A \\sin B[\/latex]<\/p>\n<\/li>\n<li data-start=\"626\" data-end=\"692\">\n<p data-start=\"628\" data-end=\"692\">[latex]\\cos(A \\pm B) = \\cos A \\cos B \\mp \\sin A \\sin B[\/latex]<\/p>\n<\/li>\n<li data-start=\"693\" data-end=\"774\">\n<p data-start=\"695\" data-end=\"774\">[latex]\\tan(A \\pm B) = \\dfrac{\\tan A \\pm \\tan B}{1 \\mp \\tan A \\tan B}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Sum and Difference Formulas<\/strong><\/p>\n<ol>\n<li data-start=\"834\" data-end=\"1079\">\n<p data-start=\"837\" data-end=\"864\"><strong data-start=\"837\" data-end=\"862\">Remember the Patterns<\/strong><\/p>\n<ul data-start=\"868\" data-end=\"1079\">\n<li data-start=\"868\" data-end=\"933\">\n<p data-start=\"870\" data-end=\"933\">Sine keeps the same sign: plus for sum, minus for difference.<\/p>\n<\/li>\n<li data-start=\"937\" data-end=\"999\">\n<p data-start=\"939\" data-end=\"999\">Cosine flips the sign: plus for difference, minus for sum.<\/p>\n<\/li>\n<li data-start=\"1003\" data-end=\"1079\">\n<p data-start=\"1005\" data-end=\"1079\">Tangent has the same sign in the numerator, opposite in the denominator.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1081\" data-end=\"1346\">\n<p data-start=\"1084\" data-end=\"1108\"><strong data-start=\"1084\" data-end=\"1106\">Use Special Angles<\/strong><\/p>\n<ul data-start=\"1112\" data-end=\"1346\">\n<li data-start=\"1112\" data-end=\"1275\">\n<p data-start=\"1114\" data-end=\"1275\">Break down unfamiliar angles into sums or differences of [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], [latex]60^\\circ[\/latex], or [latex]90^\\circ[\/latex].<\/p>\n<\/li>\n<li data-start=\"1279\" data-end=\"1346\">\n<p data-start=\"1281\" data-end=\"1346\">Example: [latex]\\cos(75^\\circ)=\\cos(45^\\circ+30^\\circ)[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1718\" data-end=\"1869\">\n<p data-start=\"1721\" data-end=\"1750\"><strong data-start=\"1721\" data-end=\"1748\">Check Signs by Quadrant<\/strong><\/p>\n<ul data-start=\"1896\" data-end=\"2020\">\n<li data-start=\"1754\" data-end=\"1869\">\n<p data-start=\"1756\" data-end=\"1869\">Always be mindful of whether the angle is in a quadrant where sine, cosine, or tangent is positive or negative.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Find the exact value of [latex]\\sin(75\u00b0)[\/latex] using a sum formula.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsum-001\">Show Solution<\/button><\/p>\n<div id=\"qsum-001\" class=\"hidden-answer\" style=\"display: none\">\nWrite [latex]75\u00b0 = 45\u00b0 + 30\u00b0[\/latex]Use the sum formula: [latex]\\sin(\\alpha + \\beta) = \\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta[\/latex][latex]\\begin{align*}  \\sin(75\u00b0) &= \\sin(45\u00b0 + 30\u00b0) \\\\  &= \\sin 45\u00b0 \\cos 30\u00b0 + \\cos 45\u00b0 \\sin 30\u00b0 \\\\  &= \\frac{\\sqrt{2}}{2} \\cdot \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{2}}{2} \\cdot \\frac{1}{2} \\\\  &= \\frac{\\sqrt{6}}{4} + \\frac{\\sqrt{2}}{4} \\\\  &= \\frac{\\sqrt{6} + \\sqrt{2}}{4}  \\end{align*}[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-beacchfg-liSgx_72bfg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/liSgx_72bfg?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-beacchfg-liSgx_72bfg\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661365&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-beacchfg-liSgx_72bfg&#38;vembed=0&#38;video_id=liSgx_72bfg&#38;video_target=tpm-plugin-beacchfg-liSgx_72bfg\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Using+sum+and+difference+formula+to+find+the+exact+value+with+cosine_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUsing sum and difference formula to find the exact value with cosine\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Verifying Identities with Sum and Difference Formulas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"82\" data-end=\"402\">Sum and difference formulas are powerful tools for proving that two trigonometric expressions are equal. When verifying identities, you often expand one side using these formulas, then simplify until it matches the other side. This process shows that the two sides are equivalent for all values where both are defined.<\/p>\n<p data-start=\"404\" data-end=\"423\">The formulas are:<\/p>\n<ul data-start=\"425\" data-end=\"640\">\n<li data-start=\"425\" data-end=\"491\">\n<p data-start=\"427\" data-end=\"491\">[latex]\\sin(A \\pm B) = \\sin A \\cos B \\pm \\cos A \\sin B[\/latex]<\/p>\n<\/li>\n<li data-start=\"492\" data-end=\"558\">\n<p data-start=\"494\" data-end=\"558\">[latex]\\cos(A \\pm B) = \\cos A \\cos B \\mp \\sin A \\sin B[\/latex]<\/p>\n<\/li>\n<li data-start=\"559\" data-end=\"640\">\n<p data-start=\"561\" data-end=\"640\">[latex]\\tan(A \\pm B) = \\dfrac{\\tan A \\pm \\tan B}{1 \\mp \\tan A \\tan B}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Sum and Difference Formulas in Verifications<\/strong><\/p>\n<ol>\n<li data-start=\"717\" data-end=\"865\">\n<p data-start=\"720\" data-end=\"748\"><strong data-start=\"720\" data-end=\"746\">Expand Using a Formula<\/strong><\/p>\n<ul data-start=\"752\" data-end=\"865\">\n<li data-start=\"752\" data-end=\"809\">\n<p data-start=\"754\" data-end=\"809\">Start with the more complicated side of the identity.<\/p>\n<\/li>\n<li data-start=\"813\" data-end=\"865\">\n<p data-start=\"815\" data-end=\"865\">Apply the appropriate sum or difference formula.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"867\" data-end=\"1037\">\n<p data-start=\"870\" data-end=\"894\"><strong data-start=\"870\" data-end=\"892\">Simplify Carefully<\/strong><\/p>\n<ul data-start=\"898\" data-end=\"1037\">\n<li data-start=\"898\" data-end=\"971\">\n<p data-start=\"900\" data-end=\"971\">Replace sine, cosine, or tangent of special angles with exact values.<\/p>\n<\/li>\n<li data-start=\"975\" data-end=\"1037\">\n<p data-start=\"977\" data-end=\"1037\">Combine like terms, simplify fractions, or cancel factors.<\/p>\n<\/li>\n<\/ul>\n<ul data-start=\"1172\" data-end=\"1334\">\n<li data-start=\"1304\" data-end=\"1334\">\n<p data-start=\"1339\" data-end=\"1372\"><strong data-start=\"1339\" data-end=\"1370\">Look for Target Expressions<\/strong><\/p>\n<ul data-start=\"1376\" data-end=\"1510\">\n<li data-start=\"1376\" data-end=\"1510\">\n<p data-start=\"1378\" data-end=\"1510\">If the identity involves a single trig function of a sum or difference, try collapsing products into one cosine, sine, or tangent.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Verify the identity: [latex]\\sin(x + \\pi) = -\\sin x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsum-002\">Show Solution<\/button><\/p>\n<div id=\"qsum-002\" class=\"hidden-answer\" style=\"display: none\">\nUse the sum formula: [latex]\\sin(\\alpha + \\beta) = \\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta[\/latex][latex]\\begin{align*}  \\sin(x + \\pi) &= \\sin x \\cos \\pi + \\cos x \\sin \\pi \\\\  &= \\sin x (-1) + \\cos x (0) \\\\  &= -\\sin x + 0 \\\\  &= -\\sin x  \\end{align*}[\/latex]The identity is verified.\n<\/div>\n<\/div>\n<\/section>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-befeagae-qMcYCXaveRM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/qMcYCXaveRM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-befeagae-qMcYCXaveRM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661366&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-befeagae-qMcYCXaveRM&#38;vembed=0&#38;video_id=qMcYCXaveRM&#38;video_target=tpm-plugin-befeagae-qMcYCXaveRM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+verify+an+identity+with+the+sum+and+difference+formulas_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to verify an identity with the sum and difference formulas\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Using sum and difference formula to find the exact 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