Question

${\tan }^{-1}\left(\sqrt{3}\right)-{\cot }^{-1}(-\sqrt{3})$ is equal to

• A. $0$
• B. $2\sqrt{3}$
• C. $–\frac{\mathrm{\pi }}{2}$
• D. $\mathrm{\pi }$

Answer 1

The correct option is C

$–\frac{\mathrm{\pi }}{2}$

Explanation for correct option

Evaluating the given expression:

Given, ${\tan }^{-1}\left(\sqrt{3}\right)-{\cot }^{-1}(-\sqrt{3})$

We know that, $\mathrm{\pi }-{\tan }^{-1}\left(\mathrm{\theta }\right)={\tan }^{-1}(-\mathrm{\theta })$

So,

$$\begin{gathered} \mathrm{\pi }-{\tan }^{-1}\left(\sqrt{3}\right)={\tan }^{-1}(-\sqrt{3}) \\ \Rightarrow {\tan }^{-1}\left(\sqrt{3}\right)=\mathrm{\pi }-{\tan }^{-1}(-\sqrt{3}) \end{gathered}$$

Therefore, we get:

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

Hence, the correct answer is Option (C).