Find the antiderivative of f(x)=cos2x/(sin^2x)*(cos^2x).

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We have the function f(x) = cos 2x /(sin x)^2*(cos x)^2

cos 2x = (cos x)^2 - ( sin x)^2

=> cos 2x /(sin x)^2*(cos x)^2

=> [(cos x)^2 - ( sin x)^2] / (sin x)^2*(cos x)^2

=> 1/ (sin x)^2 - 1/ (cos x)^2

=> (cosec x)^2 - (sec...

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We have the function f(x) = cos 2x /(sin x)^2*(cos x)^2

cos 2x = (cos x)^2 - ( sin x)^2

=> cos 2x /(sin x)^2*(cos x)^2

=> [(cos x)^2 - ( sin x)^2] / (sin x)^2*(cos x)^2

=> 1/ (sin x)^2 - 1/ (cos x)^2

=> (cosec x)^2 - (sec x)^2

So, Int [ f(x)]

=> Int [ (cosec x)^2 - (sec x)^2 dx]

=> Int [(cosec x)^2 dx] - Int [(sec x)^2 dx]

=> - tan x  - cot x + C

Therefore the required antiderivative of f(x)=cos2x/(sin^2x)*(cos^2x) = -tan x  - cot x + C

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