Question

The solution of ${\tan }^{-1}\left(x\right)+2{\cot }^{-1}\left(x\right)=\frac{2\mathrm{\pi }}{3}$ is

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

Answer 1

The correct option is D

$\sqrt{3}$

Explanation for correct option

Given: ${\tan }^{-1}\left(x\right)+2{\cot }^{-1}\left(x\right)=\frac{2\mathrm{\pi }}{3}$

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

Hence, option D is correct.