Question

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

• 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

Answer: (A)

Given that
$\tan \frac{\pi }{5}+2\tan \frac{2\pi }{5}+4\cot \frac{4\pi }{5}$
$=\tan \frac{\pi }{5}+2\left[2\cot \frac{4\pi }{5}+\tan \frac{2\pi }{5}\right]$
$=\tan \frac{\pi }{5}+2\cot \frac{2\pi }{5}$
$[\because 2\cot 2A+\tan A=\cot A]$
$=\tan \frac{\pi }{5}+\cot \frac{\pi }{5}-\tan \frac{\pi }{5}=\cot \frac{\pi }{5}$