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Question

Let f(θ)=sin(tan1(sinθcos2θ));π4<θ<π4, Then the value of d(d(tanθ))(f(θ)) is

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Solution

f(θ)=sin(tan1(sinθcos2θ))f(θ)=sin(tan1(sinθ2cos2θ1))f(θ)=sin(sin1(sinθsin2θ+2cos2θ1))f(θ)=sin(sin1(sinθcos2θ))f(θ)=sin(sin1(tanθ))=tanθ
So,
d(d(tanθ))(f(θ))=dd(tanθ)tanθ=1

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