{"id":1530,"date":"2025-07-25T02:35:46","date_gmt":"2025-07-25T02:35:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1530"},"modified":"2026-03-12T06:00:39","modified_gmt":"2026-03-12T06:00:39","slug":"simplifying-trigonometric-expressions-with-identities-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/simplifying-trigonometric-expressions-with-identities-fresh-take\/","title":{"raw":"Simplifying Trigonometric Expressions with Identities: Fresh Take","rendered":"Simplifying Trigonometric Expressions with Identities: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Verify the fundamental trigonometric identities.<\/li>\r\n \t<li>Simplify trigonometric expressions using algebra and the identities.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Verifying Fundamental Trigonometric Identities<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nVerifying trigonometric identities means showing that two different-looking expressions are actually equal for all values where both sides are defined. The key is to use the <strong data-start=\"249\" data-end=\"275\">fundamental identities<\/strong>\u2014reciprocal, quotient, and Pythagorean relationships\u2014to rewrite one side until it matches the other. This process is not about solving for a variable, but about proving equality through algebraic manipulation and trig rules.\r\n\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Verifying Identities<\/strong>\r\n<ol>\r\n \t<li data-start=\"548\" data-end=\"681\">\r\n<p data-start=\"551\" data-end=\"576\"><strong data-start=\"551\" data-end=\"574\">Start with One Side<\/strong><\/p>\r\n\r\n<ul data-start=\"580\" data-end=\"681\">\r\n \t<li data-start=\"580\" data-end=\"639\">\r\n<p data-start=\"582\" data-end=\"639\">Pick the more complicated side and work to simplify it.<\/p>\r\n<\/li>\r\n \t<li data-start=\"643\" data-end=\"681\">\r\n<p data-start=\"645\" data-end=\"681\">Avoid touching both sides at once.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"683\" data-end=\"964\">\r\n<p data-start=\"686\" data-end=\"730\"><strong data-start=\"686\" data-end=\"728\">Use Reciprocal and Quotient Identities<\/strong><\/p>\r\n\r\n<ul data-start=\"734\" data-end=\"964\">\r\n \t<li data-start=\"734\" data-end=\"838\">\r\n<p data-start=\"736\" data-end=\"838\">[latex]\\sin\\theta = \\dfrac{1}{\\csc\\theta}[\/latex], [latex]\\cos\\theta = \\dfrac{1}{\\sec\\theta}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"842\" data-end=\"964\">\r\n<p data-start=\"844\" data-end=\"964\">[latex]\\tan\\theta = \\dfrac{\\sin\\theta}{\\cos\\theta}[\/latex], [latex]\\cot\\theta = \\dfrac{\\cos\\theta}{\\sin\\theta}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"966\" data-end=\"1165\">\r\n<p data-start=\"969\" data-end=\"1003\"><strong data-start=\"969\" data-end=\"1001\">Apply Pythagorean Identities<\/strong><\/p>\r\n\r\n<ul data-start=\"1007\" data-end=\"1165\">\r\n \t<li data-start=\"1007\" data-end=\"1057\">\r\n<p data-start=\"1009\" data-end=\"1057\">[latex]\\sin^{2}\\theta+\\cos^{2}\\theta=1[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1061\" data-end=\"1111\">\r\n<p data-start=\"1063\" data-end=\"1111\">[latex]1+\\tan^{2}\\theta=\\sec^{2}\\theta[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1115\" data-end=\"1165\">\r\n<p data-start=\"1117\" data-end=\"1165\">[latex]1+\\cot^{2}\\theta=\\csc^{2}\\theta[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1167\" data-end=\"1287\">\r\n<p data-start=\"1170\" data-end=\"1194\"><strong data-start=\"1170\" data-end=\"1192\">Algebra Tools Help<\/strong><\/p>\r\n\r\n<ul data-start=\"1198\" data-end=\"1287\">\r\n \t<li data-start=\"1198\" data-end=\"1287\">\r\n<p data-start=\"1200\" data-end=\"1287\">Factor, expand, combine fractions, or multiply by conjugates to simplify expressions.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1289\" data-end=\"1444\">\r\n<p data-start=\"1292\" data-end=\"1320\"><strong data-start=\"1292\" data-end=\"1318\">Work Toward the Target<\/strong><\/p>\r\n\r\n<ul data-start=\"1324\" data-end=\"1444\">\r\n \t<li data-start=\"1324\" data-end=\"1388\">\r\n<p data-start=\"1326\" data-end=\"1388\">Always aim to make your expression look like the other side.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1392\" data-end=\"1444\">\r\n<p data-start=\"1394\" data-end=\"1444\">Once both sides match, the identity is verified.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1446\" data-end=\"1765\">\r\n<p data-start=\"1449\" data-end=\"1523\"><strong data-start=\"1449\" data-end=\"1460\">Example: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Verify [latex]\\dfrac{1}{\\sec\\theta}=\\cos\\theta[\/latex].<\/span><\/p>\r\n<\/li>\r\n<\/ol>\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li data-start=\"1527\" data-end=\"1584\">\r\n<p data-start=\"1529\" data-end=\"1584\">Start with LHS: [latex]\\dfrac{1}{\\sec\\theta}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1588\" data-end=\"1668\">\r\n<p data-start=\"1590\" data-end=\"1668\">Replace [latex]\\sec\\theta[\/latex] with [latex]\\dfrac{1}{\\cos\\theta}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1672\" data-end=\"1745\">\r\n<p data-start=\"1674\" data-end=\"1745\">Simplify: [latex]\\dfrac{1}{\\tfrac{1}{\\cos\\theta}}=\\cos\\theta[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1749\" data-end=\"1765\">\r\n<p data-start=\"1751\" data-end=\"1765\">LHS = RHS.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Verify the identity: [latex]\\tan x \\cos x = \\sin x[\/latex][reveal-answer q=\"verify-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"verify-001\"]\r\nStart with the left side and simplify:\r\n[latex]\\begin{align*}\r\n\\tan x \\cos x &amp;= \\frac{\\sin x}{\\cos x} \\cdot \\cos x &amp;&amp; \\text{use } \\tan x = \\frac{\\sin x}{\\cos x} \\\\\r\n&amp;= \\sin x &amp;&amp; \\text{simplify}\r\n\\end{align*}[\/latex]The left side equals the right side, so 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-ahdfagec-97f0ixoHtqA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/97f0ixoHtqA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ahdfagec-97f0ixoHtqA\" 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=14661358&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ahdfagec-97f0ixoHtqA&vembed=0&video_id=97f0ixoHtqA&video_target=tpm-plugin-ahdfagec-97f0ixoHtqA'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Trig+5.2+-+Verifying+Trigonometric+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTrig 5.2 - Verifying Trigonometric Identities\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Simplifying Trigonometric Expressions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"66\" data-end=\"528\">Simplifying trigonometric expressions means taking a complicated trig expression and using algebra together with the fundamental identities to rewrite it in a simpler form. The goal is not to \u201cprove\u201d equality (like with identities) but to reduce the expression so it\u2019s easier to work with. This process combines factoring, expanding, and reducing fractions with the reciprocal, quotient, and Pythagorean identities to make the expression as simple as possible.<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Simplifying Trig Expressions<\/strong>\r\n<ol data-start=\"583\" data-end=\"1875\">\r\n \t<li data-start=\"583\" data-end=\"914\">\r\n<p data-start=\"586\" data-end=\"623\"><strong data-start=\"586\" data-end=\"621\">Look for Identity Substitutions<\/strong><\/p>\r\n\r\n<ul data-start=\"627\" data-end=\"914\">\r\n \t<li data-start=\"627\" data-end=\"781\">\r\n<p data-start=\"629\" data-end=\"781\">Replace [latex]\\sec\\theta[\/latex] with [latex]\\dfrac{1}{\\cos\\theta}[\/latex], [latex]\\csc\\theta[\/latex] with [latex]\\dfrac{1}{\\sin\\theta}[\/latex], etc.<\/p>\r\n<\/li>\r\n \t<li data-start=\"785\" data-end=\"914\">\r\n<p data-start=\"787\" data-end=\"914\">Use [latex]\\tan\\theta = \\dfrac{\\sin\\theta}{\\cos\\theta}[\/latex] or [latex]\\cot\\theta = \\dfrac{\\cos\\theta}{\\sin\\theta}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"916\" data-end=\"1115\">\r\n<p data-start=\"919\" data-end=\"953\"><strong data-start=\"919\" data-end=\"951\">Apply Pythagorean Identities<\/strong><\/p>\r\n\r\n<ul data-start=\"957\" data-end=\"1115\">\r\n \t<li data-start=\"957\" data-end=\"1007\">\r\n<p data-start=\"959\" data-end=\"1007\">[latex]\\sin^{2}\\theta+\\cos^{2}\\theta=1[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1011\" data-end=\"1061\">\r\n<p data-start=\"1013\" data-end=\"1061\">[latex]1+\\tan^{2}\\theta=\\sec^{2}\\theta[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1065\" data-end=\"1115\">\r\n<p data-start=\"1067\" data-end=\"1115\">[latex]1+\\cot^{2}\\theta=\\csc^{2}\\theta[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1117\" data-end=\"1286\">\r\n<p data-start=\"1120\" data-end=\"1145\"><strong data-start=\"1120\" data-end=\"1143\">Use Algebraic Tools<\/strong><\/p>\r\n\r\n<ul data-start=\"1149\" data-end=\"1286\">\r\n \t<li data-start=\"1149\" data-end=\"1173\">\r\n<p data-start=\"1151\" data-end=\"1173\">Factor common terms.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1177\" data-end=\"1242\">\r\n<p data-start=\"1179\" data-end=\"1242\">Multiply numerator and denominator by conjugates to simplify.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1246\" data-end=\"1286\">\r\n<p data-start=\"1248\" data-end=\"1286\">Cancel common factors when possible.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1288\" data-end=\"1443\">\r\n<p data-start=\"1291\" data-end=\"1314\"><strong data-start=\"1291\" data-end=\"1312\">Work Step by Step<\/strong><\/p>\r\n\r\n<ul data-start=\"1318\" data-end=\"1443\">\r\n \t<li data-start=\"1318\" data-end=\"1366\">\r\n<p data-start=\"1320\" data-end=\"1366\">Don\u2019t try to jump to the answer in one move.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1370\" data-end=\"1443\">\r\n<p data-start=\"1372\" data-end=\"1443\">Each substitution or algebra step should make the expression simpler.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1445\" data-end=\"1875\">\r\n<p data-start=\"1448\" data-end=\"1528\"><strong data-start=\"1448\" data-end=\"1459\">Example: <\/strong>Simplify [latex]\\dfrac{\\sin^{2}\\theta}{1-\\cos\\theta}[\/latex].<\/p>\r\n\r\n<ul data-start=\"1532\" data-end=\"1875\">\r\n \t<li data-start=\"1532\" data-end=\"1607\">\r\n<p data-start=\"1534\" data-end=\"1607\">Rewrite numerator using [latex]\\sin^{2}\\theta=1-\\cos^{2}\\theta[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1611\" data-end=\"1688\">\r\n<p data-start=\"1613\" data-end=\"1688\">Expression becomes [latex]\\dfrac{1-\\cos^{2}\\theta}{1-\\cos\\theta}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1692\" data-end=\"1780\">\r\n<p data-start=\"1694\" data-end=\"1780\">Factor numerator: [latex]\\dfrac{(1-\\cos\\theta)(1+\\cos\\theta)}{1-\\cos\\theta}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1784\" data-end=\"1825\">\r\n<p data-start=\"1786\" data-end=\"1825\">Cancel [latex]1-\\cos\\theta[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1829\" data-end=\"1875\">\r\n<p data-start=\"1831\" data-end=\"1875\">Final answer: [latex]1+\\cos\\theta[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Simplify: [latex]\\frac{\\sin x}{\\csc x} + \\frac{\\cos x}{\\sec x}[\/latex][reveal-answer q=\"simplify-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"simplify-001\"]\r\nUse reciprocal identities: [latex]\\csc x = \\frac{1}{\\sin x}[\/latex] and [latex]\\sec x = \\frac{1}{\\cos x}[\/latex][latex]\\begin{align*}\r\n\\frac{\\sin x}{\\csc x} + \\frac{\\cos x}{\\sec x} &amp;= \\frac{\\sin x}{\\frac{1}{\\sin x}} + \\frac{\\cos x}{\\frac{1}{\\cos x}} \\\\\r\n&amp;= \\sin x \\cdot \\sin x + \\cos x \\cdot \\cos x \\\\\r\n&amp;= \\sin^2 x + \\cos^2 x \\\\\r\n&amp;= 1\r\n\\end{align*}[\/latex]The simplified expression is [latex]1[\/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-gcfddeba-Hf0AciRDDaE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Hf0AciRDDaE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gcfddeba-Hf0AciRDDaE\" 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=14661359&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gcfddeba-Hf0AciRDDaE&vembed=0&video_id=Hf0AciRDDaE&video_target=tpm-plugin-gcfddeba-Hf0AciRDDaE'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+1+-+Simplifying+a+Trigonometric+Expression_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Simplifying a Trigonometric Expression\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>Verify the fundamental trigonometric identities.<\/li>\n<li>Simplify trigonometric expressions using algebra and the identities.<\/li>\n<\/ul>\n<\/section>\n<h2>Verifying Fundamental Trigonometric Identities<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Verifying trigonometric identities means showing that two different-looking expressions are actually equal for all values where both sides are defined. The key is to use the <strong data-start=\"249\" data-end=\"275\">fundamental identities<\/strong>\u2014reciprocal, quotient, and Pythagorean relationships\u2014to rewrite one side until it matches the other. This process is not about solving for a variable, but about proving equality through algebraic manipulation and trig rules.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Verifying Identities<\/strong><\/p>\n<ol>\n<li data-start=\"548\" data-end=\"681\">\n<p data-start=\"551\" data-end=\"576\"><strong data-start=\"551\" data-end=\"574\">Start with One Side<\/strong><\/p>\n<ul data-start=\"580\" data-end=\"681\">\n<li data-start=\"580\" data-end=\"639\">\n<p data-start=\"582\" data-end=\"639\">Pick the more complicated side and work to simplify it.<\/p>\n<\/li>\n<li data-start=\"643\" data-end=\"681\">\n<p data-start=\"645\" data-end=\"681\">Avoid touching both sides at once.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"683\" data-end=\"964\">\n<p data-start=\"686\" data-end=\"730\"><strong data-start=\"686\" data-end=\"728\">Use Reciprocal and Quotient Identities<\/strong><\/p>\n<ul data-start=\"734\" data-end=\"964\">\n<li data-start=\"734\" data-end=\"838\">\n<p data-start=\"736\" data-end=\"838\">[latex]\\sin\\theta = \\dfrac{1}{\\csc\\theta}[\/latex], [latex]\\cos\\theta = \\dfrac{1}{\\sec\\theta}[\/latex]<\/p>\n<\/li>\n<li data-start=\"842\" data-end=\"964\">\n<p data-start=\"844\" data-end=\"964\">[latex]\\tan\\theta = \\dfrac{\\sin\\theta}{\\cos\\theta}[\/latex], [latex]\\cot\\theta = \\dfrac{\\cos\\theta}{\\sin\\theta}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"966\" data-end=\"1165\">\n<p data-start=\"969\" data-end=\"1003\"><strong data-start=\"969\" data-end=\"1001\">Apply Pythagorean Identities<\/strong><\/p>\n<ul data-start=\"1007\" data-end=\"1165\">\n<li data-start=\"1007\" data-end=\"1057\">\n<p data-start=\"1009\" data-end=\"1057\">[latex]\\sin^{2}\\theta+\\cos^{2}\\theta=1[\/latex]<\/p>\n<\/li>\n<li data-start=\"1061\" data-end=\"1111\">\n<p data-start=\"1063\" data-end=\"1111\">[latex]1+\\tan^{2}\\theta=\\sec^{2}\\theta[\/latex]<\/p>\n<\/li>\n<li data-start=\"1115\" data-end=\"1165\">\n<p data-start=\"1117\" data-end=\"1165\">[latex]1+\\cot^{2}\\theta=\\csc^{2}\\theta[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1167\" data-end=\"1287\">\n<p data-start=\"1170\" data-end=\"1194\"><strong data-start=\"1170\" data-end=\"1192\">Algebra Tools Help<\/strong><\/p>\n<ul data-start=\"1198\" data-end=\"1287\">\n<li data-start=\"1198\" data-end=\"1287\">\n<p data-start=\"1200\" data-end=\"1287\">Factor, expand, combine fractions, or multiply by conjugates to simplify expressions.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1289\" data-end=\"1444\">\n<p data-start=\"1292\" data-end=\"1320\"><strong data-start=\"1292\" data-end=\"1318\">Work Toward the Target<\/strong><\/p>\n<ul data-start=\"1324\" data-end=\"1444\">\n<li data-start=\"1324\" data-end=\"1388\">\n<p data-start=\"1326\" data-end=\"1388\">Always aim to make your expression look like the other side.<\/p>\n<\/li>\n<li data-start=\"1392\" data-end=\"1444\">\n<p data-start=\"1394\" data-end=\"1444\">Once both sides match, the identity is verified.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1446\" data-end=\"1765\">\n<p data-start=\"1449\" data-end=\"1523\"><strong data-start=\"1449\" data-end=\"1460\">Example: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Verify [latex]\\dfrac{1}{\\sec\\theta}=\\cos\\theta[\/latex].<\/span><\/p>\n<\/li>\n<\/ol>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li data-start=\"1527\" data-end=\"1584\">\n<p data-start=\"1529\" data-end=\"1584\">Start with LHS: [latex]\\dfrac{1}{\\sec\\theta}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1588\" data-end=\"1668\">\n<p data-start=\"1590\" data-end=\"1668\">Replace [latex]\\sec\\theta[\/latex] with [latex]\\dfrac{1}{\\cos\\theta}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1672\" data-end=\"1745\">\n<p data-start=\"1674\" data-end=\"1745\">Simplify: [latex]\\dfrac{1}{\\tfrac{1}{\\cos\\theta}}=\\cos\\theta[\/latex].<\/p>\n<\/li>\n<li data-start=\"1749\" data-end=\"1765\">\n<p data-start=\"1751\" data-end=\"1765\">LHS = RHS.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Verify the identity: [latex]\\tan x \\cos x = \\sin x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qverify-001\">Show Solution<\/button><\/p>\n<div id=\"qverify-001\" class=\"hidden-answer\" style=\"display: none\">\nStart with the left side and simplify:<br \/>\n[latex]\\begin{align*}  \\tan x \\cos x &= \\frac{\\sin x}{\\cos x} \\cdot \\cos x && \\text{use } \\tan x = \\frac{\\sin x}{\\cos x} \\\\  &= \\sin x && \\text{simplify}  \\end{align*}[\/latex]The left side equals the right side, so 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-ahdfagec-97f0ixoHtqA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/97f0ixoHtqA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ahdfagec-97f0ixoHtqA\" 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=14661358&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ahdfagec-97f0ixoHtqA&#38;vembed=0&#38;video_id=97f0ixoHtqA&#38;video_target=tpm-plugin-ahdfagec-97f0ixoHtqA\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Trig+5.2+-+Verifying+Trigonometric+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTrig 5.2 &#8211; Verifying Trigonometric Identities\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Simplifying Trigonometric Expressions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"66\" data-end=\"528\">Simplifying trigonometric expressions means taking a complicated trig expression and using algebra together with the fundamental identities to rewrite it in a simpler form. The goal is not to \u201cprove\u201d equality (like with identities) but to reduce the expression so it\u2019s easier to work with. This process combines factoring, expanding, and reducing fractions with the reciprocal, quotient, and Pythagorean identities to make the expression as simple as possible.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Simplifying Trig Expressions<\/strong><\/p>\n<ol data-start=\"583\" data-end=\"1875\">\n<li data-start=\"583\" data-end=\"914\">\n<p data-start=\"586\" data-end=\"623\"><strong data-start=\"586\" data-end=\"621\">Look for Identity Substitutions<\/strong><\/p>\n<ul data-start=\"627\" data-end=\"914\">\n<li data-start=\"627\" data-end=\"781\">\n<p data-start=\"629\" data-end=\"781\">Replace [latex]\\sec\\theta[\/latex] with [latex]\\dfrac{1}{\\cos\\theta}[\/latex], [latex]\\csc\\theta[\/latex] with [latex]\\dfrac{1}{\\sin\\theta}[\/latex], etc.<\/p>\n<\/li>\n<li data-start=\"785\" data-end=\"914\">\n<p data-start=\"787\" data-end=\"914\">Use [latex]\\tan\\theta = \\dfrac{\\sin\\theta}{\\cos\\theta}[\/latex] or [latex]\\cot\\theta = \\dfrac{\\cos\\theta}{\\sin\\theta}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"916\" data-end=\"1115\">\n<p data-start=\"919\" data-end=\"953\"><strong data-start=\"919\" data-end=\"951\">Apply Pythagorean Identities<\/strong><\/p>\n<ul data-start=\"957\" data-end=\"1115\">\n<li data-start=\"957\" data-end=\"1007\">\n<p data-start=\"959\" data-end=\"1007\">[latex]\\sin^{2}\\theta+\\cos^{2}\\theta=1[\/latex]<\/p>\n<\/li>\n<li data-start=\"1011\" data-end=\"1061\">\n<p data-start=\"1013\" data-end=\"1061\">[latex]1+\\tan^{2}\\theta=\\sec^{2}\\theta[\/latex]<\/p>\n<\/li>\n<li data-start=\"1065\" data-end=\"1115\">\n<p data-start=\"1067\" data-end=\"1115\">[latex]1+\\cot^{2}\\theta=\\csc^{2}\\theta[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1117\" data-end=\"1286\">\n<p data-start=\"1120\" data-end=\"1145\"><strong data-start=\"1120\" data-end=\"1143\">Use Algebraic Tools<\/strong><\/p>\n<ul data-start=\"1149\" data-end=\"1286\">\n<li data-start=\"1149\" data-end=\"1173\">\n<p data-start=\"1151\" data-end=\"1173\">Factor common terms.<\/p>\n<\/li>\n<li data-start=\"1177\" data-end=\"1242\">\n<p data-start=\"1179\" data-end=\"1242\">Multiply numerator and denominator by conjugates to simplify.<\/p>\n<\/li>\n<li data-start=\"1246\" data-end=\"1286\">\n<p data-start=\"1248\" data-end=\"1286\">Cancel common factors when possible.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1288\" data-end=\"1443\">\n<p data-start=\"1291\" data-end=\"1314\"><strong data-start=\"1291\" data-end=\"1312\">Work Step by Step<\/strong><\/p>\n<ul data-start=\"1318\" data-end=\"1443\">\n<li data-start=\"1318\" data-end=\"1366\">\n<p data-start=\"1320\" data-end=\"1366\">Don\u2019t try to jump to the answer in one move.<\/p>\n<\/li>\n<li data-start=\"1370\" data-end=\"1443\">\n<p data-start=\"1372\" data-end=\"1443\">Each substitution or algebra step should make the expression simpler.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1445\" data-end=\"1875\">\n<p data-start=\"1448\" data-end=\"1528\"><strong data-start=\"1448\" data-end=\"1459\">Example: <\/strong>Simplify [latex]\\dfrac{\\sin^{2}\\theta}{1-\\cos\\theta}[\/latex].<\/p>\n<ul data-start=\"1532\" data-end=\"1875\">\n<li data-start=\"1532\" data-end=\"1607\">\n<p data-start=\"1534\" data-end=\"1607\">Rewrite numerator using [latex]\\sin^{2}\\theta=1-\\cos^{2}\\theta[\/latex].<\/p>\n<\/li>\n<li data-start=\"1611\" data-end=\"1688\">\n<p data-start=\"1613\" data-end=\"1688\">Expression becomes [latex]\\dfrac{1-\\cos^{2}\\theta}{1-\\cos\\theta}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1692\" data-end=\"1780\">\n<p data-start=\"1694\" data-end=\"1780\">Factor numerator: [latex]\\dfrac{(1-\\cos\\theta)(1+\\cos\\theta)}{1-\\cos\\theta}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1784\" data-end=\"1825\">\n<p data-start=\"1786\" data-end=\"1825\">Cancel [latex]1-\\cos\\theta[\/latex].<\/p>\n<\/li>\n<li data-start=\"1829\" data-end=\"1875\">\n<p data-start=\"1831\" data-end=\"1875\">Final answer: [latex]1+\\cos\\theta[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Simplify: [latex]\\frac{\\sin x}{\\csc x} + \\frac{\\cos x}{\\sec x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsimplify-001\">Show Solution<\/button><\/p>\n<div id=\"qsimplify-001\" class=\"hidden-answer\" style=\"display: none\">\nUse reciprocal identities: [latex]\\csc x = \\frac{1}{\\sin x}[\/latex] and [latex]\\sec x = \\frac{1}{\\cos x}[\/latex][latex]\\begin{align*}  \\frac{\\sin x}{\\csc x} + \\frac{\\cos x}{\\sec x} &= \\frac{\\sin x}{\\frac{1}{\\sin x}} + \\frac{\\cos x}{\\frac{1}{\\cos x}} \\\\  &= \\sin x \\cdot \\sin x + \\cos x \\cdot \\cos x \\\\  &= \\sin^2 x + \\cos^2 x \\\\  &= 1  \\end{align*}[\/latex]The simplified expression is [latex]1[\/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-gcfddeba-Hf0AciRDDaE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Hf0AciRDDaE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gcfddeba-Hf0AciRDDaE\" 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=14661359&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gcfddeba-Hf0AciRDDaE&#38;vembed=0&#38;video_id=Hf0AciRDDaE&#38;video_target=tpm-plugin-gcfddeba-Hf0AciRDDaE\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+1+-+Simplifying+a+Trigonometric+Expression_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Simplifying a Trigonometric Expression\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Trig 5.2 - Verifying Trigonometric Identities\",\"author\":\"\",\"organization\":\"Mrs. 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