Question

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 $\frac{\pi }{5}$

Given

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

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

$(\cot 2\alpha =\frac{{\cot }^2\alpha -1}{2\cot \alpha })$

$\tan \left(\frac{\pi }{5}\right)+2\tan 2\left(\frac{\pi }{5}\right)+2\left(\cot 2\left(\frac{\pi }{5}\right)-\tan 2\left(\frac{\pi }{5}\right)\right)$

$\tan \left(\frac{\pi }{5}\right)+2\tan 2\left(\frac{\pi }{5}\right)+2\cot 2\left(\frac{\pi }{5}\right)-2\tan 2\left(\frac{\pi }{5}\right)$

$\tan \left(\frac{\pi }{5}\right)+2\cot 2\left(\frac{\pi }{5}\right)$

$\tan \left(\frac{\pi }{5}\right)+2\left(\frac{{\cot }^{2}2\left(\frac{\pi }{5}\right)-1}{2\cot 2\left(\frac{\pi }{5}\right)}\right)$

$(\cot 2\alpha =\frac{{\cot }^2\alpha -1}{2\cot \alpha })$

$\tan \left(\frac{\pi }{5}\right)+\left(\cot \left(\frac{\pi }{5}\right)-\tan \left(\frac{\pi }{5}\right)\right)$

$\tan \left(\frac{\pi }{5}\right)+\cot \left(\frac{\pi }{5}\right)-\tan \left(\frac{\pi }{5}\right)$

$\cot \left(\frac{\pi }{5}\right)$