Question

The value of $\tan \frac{\pi }{5}+2\tan \frac{2\pi }{5}+4\cot \frac{4\pi }{5}$ is

• A. $\cot \frac{\pi }{5}$
• B. $\cot \frac{2\pi }{5}$
• C. $\cot \frac{4\pi }{5}$
• D. $\cot \frac{3\pi }{5}$

Answer 1

The correct option is A $\cot \frac{\pi }{5}$

$\tan \frac{\pi }{5}+2\tan \frac{2\pi }{5}+4\cot \frac{4\pi }{5}$

$=\tan \frac{\pi }{5}+2\tan \frac{2\pi }{5}+4.\frac{\frac{1}{2\tan \frac{2\pi }{5}}}{1-\tan {}^2\frac{2\pi }{5}}$

$=\tan \frac{\pi }{5}+2\tan \frac{2\pi }{5}+2\frac{1-\tan {}^2\frac{2\pi }{5}}{\tan \frac{2\pi }{5}}$

$=\tan \frac{\pi }{5}+2\tan \frac{2\pi }{5}+\frac{2}{\tan \frac{2\pi }{5}}-2\tan \frac{2\pi }{5}$

$=\tan \frac{\pi }{5}+\frac{2}{\tan \frac{2\pi }{5}}$

$=\tan \frac{\pi }{5}+\frac{\frac{2}{2\tan \frac{\pi }{5}}}{1-\tan {}^2\frac{\pi }{5}}$

$=\tan \frac{\pi }{5}+\frac{1-\tan {}^2\frac{\pi }{5}}{\tan \frac{\pi }{5}}$

$=\tan \frac{\pi }{5}+\frac{1}{\tan \frac{\pi }{5}}-\tan \frac{\pi }{5}$

$=\cot \frac{\pi }{5}$

Hence, the answer is $\cot \frac{\pi }{5}.$