Question

Find the value of x,

if ${\tan }^{-1}x+2{\cot }^{-1}x=\frac{2\pi }{3}$.

Answer 1

${\tan }^{-1}x+2{\cot }^{-1}x=\frac{2\pi }{3}$

We know that ${\cot }^{-1}x=\frac{\pi }{2}-{\tan }^{-1}x$

$\Rightarrow {\tan }^{-1}x+2(\frac{\pi }{2}-{\tan }^{-1}x)=\frac{2\pi }{3}$

$\Rightarrow {\tan }^{-1}x+\pi -2{\tan }^{-1}x=\frac{2\pi }{3}$

$\Rightarrow {\tan }^{-1}x-2{\tan }^{-1}x=\frac{2\pi }{3}-\pi$

$\Rightarrow -{\tan }^{-1}x=\frac{2\pi -3\pi }{3}$

$\Rightarrow -{\tan }^{-1}x=\frac{-\pi }{3}$

$\Rightarrow {\tan }^{-1}x=\frac{\pi }{3}$

$\Rightarrow x=\tan \frac{\pi }{3}=\sqrt{3}$

$\therefore x=\sqrt{3}$