Prove that, tanA + 2tan2A + 4tan4A + 8cot8A = cotA using tanA + cotA = 2cosec 2A & cot A - tan A = 2*cot2A

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The identity tan A + 2*tan 2A + 4*tan 4A + 8*cot 8A = cot A has to be proved.

If tan A + 2*tan 2A + 4*tan 4A + 8*cot 8A = cot A

=> tan A + 2*tan 2A + 4*tan 4A + 8*cot 8A - cot A = 0

Using the identities tan A + cot A = 2*cosec 2A & cot A - tan A = 2*cot 2A

tan A + 2*tan 2A + 4*tan 4A + 8*cot 8A - cot A

=> tan A - cot A + 2*tan 2A + 4*tan 4A + 8*cot 8A

=> -2*cot 2A + 2*tan 2A + 4*tan 4A + 8*cot 8A

=> -2*2*cot 4A + 4*tan 4A + 8*cot 8A

=> -4*2*cot 8A + 8*cot 8A

=> 0

This proves that tan A + 2*tan 2A + 4*tan 4A + 8*cot 8A = cot A

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