{"id":28072,"date":"2025-06-20T09:08:20","date_gmt":"2025-06-20T09:08:20","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=28072"},"modified":"2025-06-20T09:08:22","modified_gmt":"2025-06-20T09:08:22","slug":"prove-the-following-trig-identity-1-_-sin","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/prove-the-following-trig-identity-1-_-sin\/","title":{"rendered":"Prove the following trig identity: 1 _ sin"},"content":{"rendered":"\n<p>Prove the following trig identity: 1 _ sin? x _ 2cosx cos- x cosx 1 +secx [4<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-light-green-cyan-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To prove a trigonometric identity, we aim to simplify one side of the equation and show that it equals the other. However, your expression appears to be incomplete or improperly formatted. From your message:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>1 _ sin? x _ 2cosx cos- x cosx 1 +secx [4<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p>This seems to be a corrupted or unclear identity. Based on common trigonometric identities, I will try to interpret it as:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>(1 \u2212 sin\u00b2x) \u2212 2cosx * cos(\u2212x) * cosx = 1 + secx<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p>Let\u2019s clarify the terms and rewrite the left-hand side (LHS) using known identities.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Use Pythagorean Identity<\/h3>\n\n\n\n<p>Recall that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>1 \u2212 sin\u00b2x = cos\u00b2x<\/strong><\/li>\n\n\n\n<li><strong>cos(\u2212x) = cosx<\/strong> (cosine is even)<\/li>\n<\/ul>\n\n\n\n<p>So we simplify the LHS:<\/p>\n\n\n\n<p><strong>LHS = (1 \u2212 sin\u00b2x) \u2212 2cosx * cos(\u2212x) * cosx<\/strong><br><strong>= cos\u00b2x \u2212 2cosx * cosx * cosx<\/strong><br><strong>= cos\u00b2x \u2212 2cos\u00b3x<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Factor the expression<\/h3>\n\n\n\n<p>Now factor the expression:<\/p>\n\n\n\n<p><strong>LHS = cos\u00b2x \u2212 2cos\u00b3x = cos\u00b2x(1 \u2212 2cosx)<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Try to Simplify or Compare with RHS<\/h3>\n\n\n\n<p>The RHS of your expression is <strong>1 + secx<\/strong>, where:<\/p>\n\n\n\n<p><strong>secx = 1\/cosx<\/strong><\/p>\n\n\n\n<p>So,<\/p>\n\n\n\n<p><strong>RHS = 1 + 1\/cosx<\/strong><\/p>\n\n\n\n<p>This is not obviously equal to <strong>cos\u00b2x(1 \u2212 2cosx)<\/strong>. That suggests either:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>There was a mistake in transcribing the original identity, or<\/li>\n\n\n\n<li>The two sides of the equation are not equal, hence not an identity.<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Final Notes<\/h3>\n\n\n\n<p>Based on what you provided, the left-hand side simplifies to <strong>cos\u00b2x(1 \u2212 2cosx)<\/strong>, while the right-hand side simplifies to <strong>1 + 1\/cosx<\/strong>. These expressions are not algebraically equivalent, so <strong>as written<\/strong>, the identity <strong>cannot be proven<\/strong> because the two sides are not equal.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-93.jpeg\" alt=\"\" class=\"wp-image-28080\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-93.jpeg 1024w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-93-300x300.jpeg 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-93-150x150.jpeg 150w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-93-768x768.jpeg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Prove the following trig identity: 1 _ sin? x _ 2cosx cos- x cosx 1 +secx [4 The Correct Answer and Explanation is: To prove a trigonometric identity, we aim to simplify one side of the equation and show that it equals the other. However, your expression appears to be incomplete or improperly formatted. From [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-28072","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/28072","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=28072"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/28072\/revisions"}],"predecessor-version":[{"id":28081,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/28072\/revisions\/28081"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=28072"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=28072"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=28072"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}