{"id":7893,"date":"2023-04-15T22:00:00","date_gmt":"2023-04-15T16:30:00","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7893"},"modified":"2023-04-10T18:21:11","modified_gmt":"2023-04-10T12:51:11","slug":"cos-2x-identity-proof-derivative-and-graph","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/cos-2x-identity-proof-derivative-and-graph\/","title":{"rendered":"Cos 2x : Identity, Proof , Derivative and Graph"},"content":{"rendered":"\n<p>Trigonometry is an essential branch of mathematics and is included in the CBSE board curriculum. The cosine function is one of the six trigonometric functions and is often used to describe periodic phenomena. In this article, we will provide an overview of cos 2x, its properties, and its applications.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Cos 2x Formula<\/h2>\n\n\n\n<p>The formula of cos 2x is given by<\/p>\n\n\n\n<p>$\\cos 2x= cos^2 x &#8211; sin^2 x$  -(1)<br>$\\cos 2x = 1 &#8211; 2 sin^2 x$  -(2)<br>$\\cos 2x= 2 cos^2x &#8211; 1$  -(3)<br>$\\cos 2x= \\frac {1 &#8211; tan^2x }{1 + tan^2 x}$  -(4)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Proof<\/h2>\n\n\n\n<p>We know that<\/p>\n\n\n\n<p>$cos (A + B) = cos A cos B  &#8211; sin A sin B$<\/p>\n\n\n\n<p>Therefore<\/p>\n\n\n\n<p>$\\cos 2x= cos (x +x) = cos (x) cos (x) &#8211; sin (x) sin (x)= cos^2x &#8211; sin^2x$  -(1)<\/p>\n\n\n\n<p>Now we know that sin^2x = 1 -cos^2x<\/p>\n\n\n\n<p>So, <br>$\\cos 2x=  cos^2x &#8211; sin^2x= cos^2 x &#8211; (1-cos^2 x) = 2cos^2 x -1$  &#8211; (3)<\/p>\n\n\n\n<p>Now we know that cos^2x = 1 -sin^2x<\/p>\n\n\n\n<p>$\\cos 2x= 2cos^2 x -1 = 1 -2 sin^2 x$  &#8211; (2)<\/p>\n\n\n\n<p>Now<\/p>\n\n\n\n<p>$\\cos 2x=  cos^2x &#8211; sin^2x = \\frac {cos^2x &#8211; sin^2x}{1} =  \\frac {cos^2x &#8211; sin^2x}{cos^2x + sin^2 x} $<br>dividing by $cos^2x$ on numerator and denominator, we get<\/p>\n\n\n\n<p>$\\cos 2x= \\frac {1 &#8211; tan^2x }{1 + tan^2 x}$  -(4)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Derivative of cos 2x<\/h2>\n\n\n\n<p>Lets find out the derivative of cos 2x<\/p>\n\n\n\n<p>$\\frac {d}{dx} cos 2x =?$<\/p>\n\n\n\n<p>Using chain rule<\/p>\n\n\n\n<p>$\\frac {d}{dx} cos 2x =-sin 2x \\times 2 = -2 sin 2x$<\/p>\n\n\n\n<p>Another method will be<\/p>\n\n\n\n<p>$\\frac {d}{dx} (cos 2x) = \\frac {d}{dx} (cos<sup>2<\/sup>x &#8211; sin<sup>2<\/sup>x)$<\/p>\n\n\n\n<p>$= \\frac {d}{dx} (cos<sup>2<\/sup>x) &#8211; \\frac {d}{dx} (sin<sup>2<\/sup>x)$<\/p>\n\n\n\n<p>$= 2  cos x \\times  (-sin x) &#8211; 2  sin x \\times  (cos x)$<\/p>\n\n\n\n<p>= -4 cos x sin x<\/p>\n\n\n\n<p>= -2 sin 2x<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Graph of Cos 2x<\/h2>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/04\/graph-of-cos2x.png\"><img loading=\"lazy\" decoding=\"async\" width=\"662\" height=\"273\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/04\/graph-of-cos2x.png\" alt=\"\" class=\"wp-image-7894\" srcset=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/04\/graph-of-cos2x.png 662w, https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/04\/graph-of-cos2x-300x124.png 300w\" sizes=\"auto, (max-width: 662px) 100vw, 662px\" \/><\/a><\/figure>\n\n\n\n<p>The graph of cos 2x is similar to cos x but  graph get shrink<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Domain and Range  of Cos 2x<\/h2>\n\n\n\n<p>Domain  and range of cos 2x is same as cos (x)<\/p>\n\n\n\n<p>Domain = $(-\\infty, \\infty)$<br>Range = $[-1, 1]$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">key properties of cos 2x<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It is a periodic function with a period of $\\pi$<\/li>\n\n\n\n<li>It is an even function, which means that cos 2(-x) = cos 2x.<\/li>\n\n\n\n<li>Its maximum value is 1, which occurs at x = 0 and x = $\\pi$.<\/li>\n\n\n\n<li>Its minimum value is -1, which occurs at x = $\\pi$\/2 and x = $3\\pi\/2$.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Trigonometry is an essential branch of mathematics and is included in the CBSE board curriculum. The cosine function is one of the six trigonometric functions and is often used to describe periodic phenomena. In this article, we will provide an overview of cos 2x, its properties, and its applications. Cos 2x Formula The formula of [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-7893","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Cos 2x : Identity, Proof , Derivative and Graph - physicscatalyst&#039;s Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/cos-2x-identity-proof-derivative-and-graph\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cos 2x : Identity, Proof , Derivative and Graph - physicscatalyst&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"Trigonometry is an essential branch of mathematics and is included in the CBSE board curriculum. 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The cosine function is one of the six trigonometric functions and is often used to describe periodic phenomena. In this article, we will provide an overview of cos 2x, its properties, and its applications. Cos 2x Formula The formula of&hellip;","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7893","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=7893"}],"version-history":[{"count":1,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7893\/revisions"}],"predecessor-version":[{"id":7895,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7893\/revisions\/7895"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=7893"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=7893"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=7893"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}