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Question

If cos2αsin2α=tan2β, then show that tan2α=cos2βsin2β.

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Solution

As we know that, 1+tan2θ=sec2θ

So, we can write
cos2αsin2α=sec2β1sec2β=cos2αsin2α+1cos2β=1cos2αsin2α+(sin2α+cos2α) (as sin2α+cos2α=1)cos2β=12cos2α

also, sin2β=1cos2β=112cos2α

So, cos2βsin2β=12cos2α(112cos2α)
=12cos2α1+12cos2α=1cos2α1=sec2α1=tan2α

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