Question

prove that:
tana + 2tan2a + 4tan4a + 8cot8a = cota

Answer 1

Solving from right

tan A + 2tan2A + 4tan4A + 8cot8A = tan A + 2tan2A + 4tan4A + 8/tan8A

= tan A + 2tan2A + 4tan4A + 8(1-tan24A)/2tan4A

= tan A + 2tan2A + [{4tan4A(tan4A)} + 4 (1-tan24A)]/tan4A

= tan A + 2tan2A + [4tan24A + 4 - tan24A]/tan4A

= tan A + 2tan2A + 4/tan4A

Proceeding as above, we will reach to

tan A + 2tan2A + 4tan4A + 8cot8A = 1/tanA = cotA

Hence proved.

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