Question

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

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

Answer 1

The correct option is C

$\sqrt{3}$

Explanation for the correct option:

Solve the equation

$\begin{array}{c}{\tan }^{-1}x+2{\cot }^{-1}x=\frac{2\pi }{3}\\ {\tan }^{-1}x+2\left(\frac{\pi }{2}-{\tan }^{-1}x\right)=\frac{2\pi }{3}\\ -{\tan }^{-1}x=\left(\frac{2\pi }{3}\right)-\pi \\ -{\tan }^{-1}x=-\frac{\pi }{3}\\ {\tan }^{-1}x=\frac{\pi }{3}\\ x=\tan \left(\frac{\pi }{3}\right)\\ =\sqrt{3}\end{array}$

Hence option (3) is correct