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Question

If sin12a1+a2cos11b21+b2=tan12x1x2, then prove that x=ab1+ab

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Solution

Let a=tanθ1,b=tanθ2 and x=tanθ

sin1(2a1+a2)cos1(1b21+b2)=tan1(2x1x2)

sin1(2tanθ11+tan2θ1)cos1(1tan2θ21+tan2θ2)=tan1(2tanθ1tan2θ)

sin1(sin2θ1)cos1(cos2θ2)=tan1(tan2θ)

2θ12θ2=2θ

θ1θ2=θ

tan(θ1θ2)=tanθ

tanθ1tanθ21+tanθ1tanθ2=tanθ

ab1+ab=tanθ

ab1+ab=x

x=ab1+ab where x=tanθ

Hence proved.

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