{"id":6823,"date":"2021-10-21T19:14:30","date_gmt":"2021-10-21T13:44:30","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6823"},"modified":"2021-10-25T00:23:32","modified_gmt":"2021-10-24T18:53:32","slug":"solve-the-equation-2tan-12x1-cos-1x","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/solve-the-equation-2tan-12x1-cos-1x\/","title":{"rendered":"Solve the equation : 2\\(tan^{-1}({2x+1})\\) = \\(cos^{-1}x\\)"},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>Here, 2\\(tan^{-1}({2x+1})\\) = \\(cos^{-1}x\\)<\/p>\n<p>cos(2\\(tan^{-1}({2x+1})\\)) = x\u00a0 { We Know cos2x = \\({1-tan^2x\\over {1+tan^2x}}\\)}<\/p>\n<p>\\(\\therefore\\)\u00a0 \\({{1-{(2x+1)}^2}\\over {1-{(2x+1)}^2}}\\) = x\u00a0 \u00a0\\(\\implies\\)\u00a0 \u00a0(1 &#8211; 2x &#8211; 1)(1 + 2x + 1) = x(\\(4x^2 + 4x + 2\\))<\/p>\n<p>\\(\\implies\\)\u00a0 -2x.2(x + 1) = 2x(\\(2x^2 + 2x + 1\\))\u00a0 \\(\\implies\\)\u00a0 2x(\\(2x^2 + 2x + 1 + 2x + 2\\)) = 0<\/p>\n<p>\\(\\implies\\) x = 0\u00a0 or \\(2x^2 + 4x + 3\\) = 0 { No Solution }<\/p>\n<p>Verify\u00a0 x = 0<\/p>\n<p>\\(2tan^{-1}(1)\\) = \\(cos^{-1}(1)\\)\u00a0 \\(\\implies\\)\u00a0 \\(\\pi\\over 2\\) = \\(\\pi\\over 2\\)<\/p>\n<p>\\(\\therefore\\)\u00a0 x = 0 is only the solution.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\">Similar Questions<\/h3>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/the-value-of-tan-11-cos-1-1over-2-sin-1-1over-2-is-equal-to\/\" aria-label=\"\u201cThe value of \\(tan^{-1}(1)\\) +  \\(cos^{-1}({-1\\over 2})\\) +  \\(sin^{-1}({-1\\over 2})\\) is equal to\u201d (Edit)\">The value of \\(tan^{-1}(1)\\) + \\(cos^{-1}({-1\\over 2})\\) + \\(sin^{-1}({-1\\over 2})\\) is equal to<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/prove-that-sin-112over-13-cot-14over-3-tan-163over-16-pi\/\" aria-label=\"\u201cProve that : \\(sin^{-1}{12\\over 13}\\) + \\(cot^{-1}{4\\over 3}\\) + \\(tan^{-1}{63\\over 16}\\) = \\(\\pi\\)\u201d (Edit)\">Prove that : \\(sin^{-1}{12\\over 13}\\) + \\(cot^{-1}{4\\over 3}\\) + \\(tan^{-1}{63\\over 16}\\) = \\(\\pi\\)<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/evaluate-sin-1sin10\/\" aria-label=\"\u201cEvaluate \\(sin^{-1}(sin10)\\)\u201d (Edit)\">Evaluate \\(sin^{-1}(sin10)\\)<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/prove-that-cos-112over-13-sin-13over-5-sin-156over-65\/\" aria-label=\"\u201cProve that : \\(cos^{-1}{12\\over 13}\\) + \\(sin^{-1}{3\\over 5}\\) = \\(sin^{-1}{56\\over 65}\\)\u201d (Edit)\">Prove that : \\(cos^{-1}{12\\over 13}\\) + \\(sin^{-1}{3\\over 5}\\) = \\(sin^{-1}{56\\over 65}\\)<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/find-the-value-of-sin-1-sqrt3over-2-cos-1cos7piover-6\/\" aria-label=\"\u201cFind the value of \\(sin^{-1}({-\\sqrt{3}\\over 2})\\) + \\(cos^{-1}(cos({7\\pi\\over 6}))\\).\u201d (Edit)\">Find the value of \\(sin^{-1}({-\\sqrt{3}\\over 2})\\) + \\(cos^{-1}(cos({7\\pi\\over 6}))\\).<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : Here, 2\\(tan^{-1}({2x+1})\\) = \\(cos^{-1}x\\) cos(2\\(tan^{-1}({2x+1})\\)) = x\u00a0 { We Know cos2x = \\({1-tan^2x\\over {1+tan^2x}}\\)} \\(\\therefore\\)\u00a0 \\({{1-{(2x+1)}^2}\\over {1-{(2x+1)}^2}}\\) = x\u00a0 \u00a0\\(\\implies\\)\u00a0 \u00a0(1 &#8211; 2x &#8211; 1)(1 + 2x + 1) = x(\\(4x^2 + 4x + 2\\)) \\(\\implies\\)\u00a0 -2x.2(x + 1) = 2x(\\(2x^2 + 2x + 1\\))\u00a0 \\(\\implies\\)\u00a0 2x(\\(2x^2 + 2x + 1 + 2x &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/solve-the-equation-2tan-12x1-cos-1x\/\"> <span class=\"screen-reader-text\">Solve the equation : 2\\(tan^{-1}({2x+1})\\) = \\(cos^{-1}x\\)<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[53,43],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Solve the equation : 2\\(tan^{-1}({2x+1})\\) = \\(cos^{-1}x\\)<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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