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

• Correct answer
  A

  $\cot \frac{\pi}{5}$
• B

  $\cot \frac{2\pi}{5}$
• C

  $\cot \frac{3\pi}{5}$
• D

  $\cot \frac{4\pi}{5}$

Solution

$\begin{aligned} & \text { (a) Given, } \tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5} \\ & =\tan \frac{\pi}{5}+2\left[2 \cot \frac{4 \pi}{5}+\tan \frac{2 \pi}{5}\right] \end{aligned}$ We know that,
$\begin{array}{ll}=\tan \frac{\pi}{5}+2 \cot \frac{2 \pi}{5} & \text { [from Eq. (i) }]\end{array}$ $=\cot \frac{\pi}{5}$ [from Eq. (i)]

Hence, the correct answer is option: 'A'.