Question

$tan\alpha +2tan2\alpha +4tan4\alpha +8cot8\alpha$=

• A. $\tan \alpha$
• B. $\tan 2\alpha$
• C. $\cot \alpha$
• D. $\cot 2\alpha$

Answer 1

The correct option is C $\cot \alpha$

$\tan \theta +2\cot 2\theta =\tan \theta +\frac{2}{\tan 2\theta }=\tan \theta +2\times \frac{1-{\tan }^2\theta }{2\tan \theta }=\tan \theta +\frac{2(1-{\tan }^2\theta )}{2\tan \theta }=\frac{2{\tan }^2\theta +2-2{\tan }^2\theta }{2\tan \theta }=\frac{2}{\tan \theta }=\cot \theta \Rightarrow \tan \theta +2\cot 2\theta =\cot \theta ...\left(1\right)\therefore \tan \alpha +2\tan 2\alpha +4\tan 4\alpha +8\cot 8\alpha =\tan \alpha +2\tan 2\alpha +4[\tan 4\alpha +2\cot 8\alpha ]=\tan \alpha +2\tan 2\alpha +4\cot 4\alpha ...[\text{using (1)]}=\tan \alpha +2[\tan 2\alpha +2\cot 4\alpha ]=\tan \alpha +2\cot 2\alpha ...[\text{using (1)]}=\cot \alpha ...[\text{using (1)]}$