Question

Solve the equation:

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

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

Answer 1

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

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

$=\tan \frac{\pi }{5}+2\tan \frac{2\pi }{5}+4\frac{1}{\tan 2\left(\frac{2\pi }{5}\right)}$

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

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

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

$=\tan \frac{\pi }{5}+2\left[\frac{1-{\tan }^2\frac{\pi }{5}}{2\tan \frac{\pi }{5}}\right]$

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

$=\cot \frac{\pi }{5}$ is the correct answer.