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Question

Prove cotαtanα2tan2α4tan4α=8cot8α.

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Solution

cotθtanθ=cos2θsinθcosθ=2cos2θsin2θ
cotαtanα2tan2α=2×cos2αsin2α2sin2αcos2α,
cotαtanα2tan2α=2×cos4αsin2α×cos2α=4cos4αsin4α
cotαtanα2tan2α4tan4α=4cos4αsin44sin4αcos4α
cotαtanα2tan2α4tan4α=4cos8αsin4cos4α=8cos8αsin8α
cotαtanα2tan2α4tan4α=8cot8α

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