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Question

tanα+2tan2α+4tan4α+8cot8α= 
  1. tanα
  2. tan2α
  3. cotα
  4. cot2α


Solution

The correct option is C cotα
tanα+2tan2α+4tan4α+8cot8α
=tanα+2tan2α+4[sin4αcos4α+2cos8αsin8α]

=tanα+2tan2α+4[cos4αcos8α+sin4αsin8α+cos4αcos8αsin8αcos4α]

=tanα+2tan2α+4[cos4α+cos4αcos8αsin8αcos4α]

=tanα+2tan2α+4[cos4α(1+cos8α)cos4αsin8α]

=tanα+2tan2α+4[2cos24α2sin4αcos4α]

=tanα+2tan2α+4cot4α

=tanα+2(tan2α+2cot4α)

=tanα+2[sin2αcos2α+2cos4αsin4α]

=tanα+2[cos2α(1+cos4α)sin4αcos2α]

=tanα+2cot2α=sinαcosα+2cos2αsin2α

=cosα+cosαcos2αsin2αcosα

=1+cos2αsin2α=2cos2α2sinαcosα=cotα

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