Question

${\tan }^{-1}x+2{\cot }^{-1}x=\frac{2\pi }{3}$, then the value of $x$ is-

• A. $\sqrt{2}$
• B. $3$
• C. $\sqrt{3}$
• D. $\frac{\sqrt{3}-1}{\sqrt{3}+1}$

Answer 1

The correct option is C $\sqrt{3}$

Since, ${\tan }^{-1}\theta +{\cot }^{-1}\theta =\frac{\pi }{2}$
$\therefore {\tan }^{-1}x+2{\cot }^{-1}x=\frac{2\pi }{3}\Rightarrow {\cot }^{-1}x+\frac{\pi }{2}=\frac{2\pi }{3}$
$\Rightarrow {\cot }^{-1}x=\frac{2\pi }{3}-\frac{\pi }{2}=\frac{\pi }{6}$

$\Rightarrow x=\sqrt{3}$