Mark the tick against the correct answer in the following:

To Find: The value of $\cot ^{-1}\left(\cot \left(\frac{5 \pi}{4}\right)\right)$

Now, let $x=\cot ^{-1}\left(\cot \left(\frac{5 \pi}{4}\right)\right)$

$\Rightarrow \cot x=\cot \left(\frac{5 \pi}{4}\right)$

Here range of principle value of $\cot$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

$\Rightarrow \mathrm{x}=\frac{5 \pi}{4} \notin\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

Hence for all values of $x$ in range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$,the value of

$\cot ^{-1}\left(\cot \left(\frac{5 \pi}{4}\right)\right)$ is

$\Rightarrow \cot x=\cot \left(\pi+\frac{\pi}{4}\right)\left(\because \cot \left(\frac{5 \pi}{4}\right)=\cot \left(\pi+\frac{\pi}{4}\right)\right)$

$\Rightarrow \cot x=\cot \left(\frac{\pi}{4}\right)(\because \cot (\pi+\theta)=\cot \theta)$

$\Rightarrow x=\frac{\pi}{4}$