Question

Find the value of $tan\frac{1}{3}(ta{n}^{-1}\pi +ta{n}^{-1}\frac{1}{\pi })$

Answer 1

$\tan \frac{1}{3}({\tan }^{-1}\pi +{\tan }^{-1}\frac{1}{\pi })⟶\left(1\right)$

We know that

${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\left(\frac{x+y}{1-xy}\right)⟶\left(2\right)$

Using $\left(2\right)$ we can rewrite $\left(1\right)$ as

$=\tan \frac{1}{3}\left({\tan }^{-1}\left(\frac{\pi +\frac{1}{\pi }}{1-\pi .\frac{1}{\pi }}\right)\right)$

$=\tan \frac{1}{3}\left({\tan }^{-1}\infty \right)$

$=\tan \frac{1}{3}\left(\frac{\pi }{2}\right)$ $(\because \tan \frac{\pi }{2}=\infty )$

$=\tan \frac{\pi }{6}$

$=\frac{1}{\sqrt{3}}$