Question

Prove that Cot A - Tan A = 4 Cot 4A + 2 Tan A

Answer 1

first of all, we find one important results .

     \begin{lgathered}cotA - tanA = \frac{1}{tanA} - tanA \\ = \frac{1-tan^2A}{tanA} \\ = \frac{2(1-tan^2A)}{2tanA} \\= 2cot2A\end{lgathered}cotA−tanA=tanA1​−tanA=tanA1−tan2A​=2tanA2(1−tan2A)​=2cot2A​  

hence,  

cotA - tanA  = 2cot2A

cotA = 2cot2A + tanA use this application here,

so,  

tanA + 2tan2A + 4{tan4A +2cot8A}

= tanA + 2tan2A + 4cot4A  by using above application  

= tanA + 2{tan2A + 2cot4A}

= tanA + 2cot2A by using above application

=cotA by using above application

hence,

tan A + 2 tan 2A + 4 tan 4A + 8 cot 8A = cotA