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Question

If tan(π4+y2) = tan3(π4+x2), prove that - siny=sinx.3+sin2x1+3sin2x.

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Solution

We know that tan(π4+α)=1+tanα1tanα=cosα+sinαcosαsinα
tan2(π4+α)=1+sin2α1sin2α2sinαcosα=sin2α
Squaring the given relation, we have to prove that
1+siny1siny=(1+sinx)3(1sinx)3.Apply comp. and divi.
2siny2=2(3sinx+sin3x)2(1+3sin2x)
siny=sinx(3+sin2x)1+3sin2x

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