Solve: ^ -1 x+2 ^ -1 x=2 3 - Vidyadip

Question

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

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Answer

$\tan^{-1}\text{x}+2\cot^{-1}\text{x}=\frac{2\pi}{3}$
$\Rightarrow\tan^{-1}\text{x}+2\Big(\frac{\pi}{2}-\tan^{-1}\text{x}\Big)=\frac{2\pi}{3}$
$\Big[\because\ \cot^{-1}\text{x}=\frac{\pi}{2}-\tan^{-1}\text{x}\Big]$
$\Rightarrow\tan^{-1}\text{x}+\pi-2\tan^{-1}\text{x}=\frac{2\pi}{3}$
$\Rightarrow\tan^{-1}\text{x}=\frac{\pi}{3}$
$\Rightarrow\tan^{-1}\text{x}=\frac{\pi}{3}$
$\Rightarrow\text{x}=\tan\frac{\pi}{3}=\sqrt3$

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