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