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Q)

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

$\begin{array}{1 1}(a)\;-\large\frac{1}{\sqrt 3}&(b)\;\large\frac{1}{\sqrt 3}\\(c)\;-\sqrt 3&(d)\;\sqrt 3\end{array}$

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A)
We have $\tan^{-1}x+2\cot^{-1}x=\large\frac{2\pi}{3}$
$\large\frac{\pi}{2}$$+\cot^{-1}x=\large\frac{2\pi}{3}$
$\Rightarrow \cot^{-1}x=\large\frac{\pi}{6}$
$x=\cot\large\frac{\pi}{6}$
$x=\sqrt 3$
Hence (d) is the correct option.
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