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Question

Prove that:
tanα+2tan2α+4tan4α+8cot8α=cotα

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Solution

tanα+2tan2α+4tanα+8cot8α=cotα

Let tanθ+2cot2θ

=sinθcosθ+2cos2θsin2θ

=sinθcosθ+2(12sin2θ)2sinθcosθ

=sin2θ+12sin2θsinθcosθ

=1sin2θsinθcosθ=cos2θsinθcosθ=cosθsinθ=cotθ

=tanα+2tan2α+4tan4α+8cot8α

=tanα+2tan2α+4(tan4α+2cot2(4α))

=tanα+2tan2α+4cot4α

=tanα+2(tan2α+2cot2(2α))

=tanα+2cot2α

=cotα=RHS.

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