if the

Disclaimer: The solution has been provided for the following question.

If $\tan ^{-1} x=\frac{\pi}{10}$ for some $x \in \mathrm{R}$, then the value of $\cot ^{-1} x$ is

(a) $\frac{\pi}{5}$

(b) $\frac{2 \pi}{5}$

(c) $\frac{3 \pi}{5}$

(d) $\frac{4 \pi}{5}$

Solution:

We know

$\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, \forall x \in R$

$\Rightarrow \frac{\pi}{10}+\cot ^{-1} x=\frac{\pi}{2}$                      $\left(\tan ^{-1} x=\frac{\pi}{10}\right)$

$\Rightarrow \cot ^{-1} x=\frac{\pi}{2}-\frac{\pi}{10}$

$\Rightarrow \cot ^{-1} x=\frac{4 \pi}{10}=\frac{2 \pi}{5}$

Hence, the correct answer is option (b).