Question:
$\tan ^{-1} x+2 \cot ^{-1} x=\frac{2 \pi}{3}$
Solution:
$\tan ^{-1} x+2 \cot ^{-1} x=\frac{2 \pi}{3}$
$\Rightarrow \tan ^{-1} x+2\left(\frac{\pi}{2}-\tan ^{-1} x\right)=\frac{2 \pi}{3}$ $\left[\because \cot ^{-1} x=\frac{\pi}{2}-\tan ^{-1} x\right]$
$\Rightarrow \tan ^{-1} x+\pi-2 \tan ^{-1} x=\frac{2 \pi}{3}$
$\Rightarrow \tan ^{-1} x=\frac{\pi}{3}$
$\Rightarrow \tan ^{-1} x=\frac{\pi}{3}$
$\Rightarrow \mathrm{x}=\tan \frac{\pi}{3}=\sqrt{3}$
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