tan^2a-cot^2a = sec^2a (1-cot^2a) prove
Mathematics · Middle School · Thu Feb 04 2021
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We have to prove: tan^2a-cot^2a = sec^2a (1-cot^2a)
sin^2a/cos^2a - cos^2a/sin^a = 1/cos^2a(1-cos^2a/sin^a) { tan^2a = sin^2a/cos^2a & cot^2a = cos^2a/sin^a}
Now, taking L.C.M.
sin^4a-cos^4a/cos^2a*sin^2a = 1/cos^2a - cos^2a/sin^2a*cos^2a
(sin^2a+cos^2a)(sin^2a-cos^2a)/cos^2a*sin^2a = sin^a - cos^2a/cos^2a*sin^2a
(1)(sin^2a-cos^2a)/cos^2a*sin^2a= sin^a - cos^2a/cos^2a*sin^2a
sin^a - cos^2a/cos^2a*sin^2a = sin^a - cos^2a/cos^2a*sin^2a
Hence proved .