{"id":40415,"date":"2025-06-28T07:32:05","date_gmt":"2025-06-28T07:32:05","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=40415"},"modified":"2025-06-28T07:32:07","modified_gmt":"2025-06-28T07:32:07","slug":"verify-the-identity-tan-2x-sec2x-1-2tan2x","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/verify-the-identity-tan-2x-sec2x-1-2tan2x\/","title":{"rendered":"Verify the identity: tan 2x + sec^2x = 1 + 2tan^2x"},"content":{"rendered":"\n<p>Verify the identity: tan 2x + sec^2x = 1 + 2tan^2x Which of the following four statements establishes the identity? A. tan^2x + sec^2x = tan^2x + (1 &#8211; tan^2x) = 1 + 2tan^2x B. tan^2x + sec^2x = tan^2x + tan^2x = 1 + 2tan^2x C. tan^2x + sec^2x = tan^2x + (tan^2x &#8211; 1) = 1 + 2tan^2x D. tan^2x + sec^2x = 1 = 1 + 2tan^2x<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"469\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-416.png\" alt=\"\" class=\"wp-image-40424\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-416.png 700w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-416-300x201.png 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The correct answer is <strong>Statement B<\/strong>.<\/p>\n\n\n\n<p>We are asked to verify the identity:<\/p>\n\n\n\n<p>tan\u20612x+sec\u20612x=1+2tan\u20612x\\tan^2 x + \\sec^2 x = 1 + 2 \\tan^2 x<\/p>\n\n\n\n<p>To do this, we recall the fundamental Pythagorean identity involving tangent and secant:<\/p>\n\n\n\n<p>sec\u20612x=1+tan\u20612x\\sec^2 x = 1 + \\tan^2 x<\/p>\n\n\n\n<p>Substituting this identity into the left-hand side of the equation:<\/p>\n\n\n\n<p>tan\u20612x+sec\u20612x=tan\u20612x+(1+tan\u20612x)\\tan^2 x + \\sec^2 x = \\tan^2 x + (1 + \\tan^2 x)<\/p>\n\n\n\n<p>Simplifying:<\/p>\n\n\n\n<p>tan\u20612x+1+tan\u20612x=1+2tan\u20612x\\tan^2 x + 1 + \\tan^2 x = 1 + 2 \\tan^2 x<\/p>\n\n\n\n<p>This confirms that the identity is true. Now we examine the four options to determine which one demonstrates this correctly.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Option A<\/strong> rewrites sec\u00b2x as 1\u2212tan\u20612&#215;1 &#8211; \\tan^2 x, which is incorrect. There is no valid identity where sec\u00b2x equals 1 minus tan\u00b2x.<\/li>\n\n\n\n<li><strong>Option B<\/strong> rewrites sec\u00b2x as 1+tan\u20612&#215;1 + \\tan^2 x, which is accurate. It substitutes this correctly and simplifies it to 1+2tan\u20612&#215;1 + 2 \\tan^2 x.<\/li>\n\n\n\n<li><strong>Option C<\/strong> uses sec\u20612x=tan\u20612x\u22121\\sec^2 x = \\tan^2 x &#8211; 1, which is also incorrect and violates the Pythagorean identity.<\/li>\n\n\n\n<li><strong>Option D<\/strong> states that tan\u20612x+sec\u20612x=1\\tan^2 x + \\sec^2 x = 1, which contradicts the original identity and does not follow logically.<\/li>\n<\/ul>\n\n\n\n<p>Therefore, <strong>Statement B is correct<\/strong> because it applies the identity sec\u20612x=1+tan\u20612x\\sec^2 x = 1 + \\tan^2 x and correctly shows how the expression simplifies to match the right-hand side.<\/p>\n\n\n\n<p>This identity is a good example of how recognizing fundamental trigonometric identities can help simplify more complex expressions. It emphasizes the importance of substitution and step-by-step simplification in proving identities.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1261.jpeg\" alt=\"\" class=\"wp-image-40434\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1261.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1261-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1261-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Verify the identity: tan 2x + sec^2x = 1 + 2tan^2x Which of the following four statements establishes the identity? A. tan^2x + sec^2x = tan^2x + (1 &#8211; tan^2x) = 1 + 2tan^2x B. tan^2x + sec^2x = tan^2x + tan^2x = 1 + 2tan^2x C. tan^2x + sec^2x = tan^2x + (tan^2x &#8211; [&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-40415","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40415","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=40415"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40415\/revisions"}],"predecessor-version":[{"id":41184,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40415\/revisions\/41184"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=40415"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=40415"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=40415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}