Mark the tick against the correct answer in the following: The value of $\cot ^{-1}\left(\cot \frac{5 \pi}{4}\right)$ is A. $\frac{\pi}{4}$ B. $\frac{-\pi}{4}$ C. $\frac{3 \pi}{4}$ D. none of these

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Mark the tick against the correct answer in the following: The value of $\cot ^{-1}\left(\cot \frac{5 \pi}{4}\right)$ is
A. $\frac{\pi}{4}$
B. $\frac{-\pi}{4}$
C. $\frac{3 \pi}{4}$
D. none of these

CBSE | Class 12 | Inverse Trigonometric Functions | Maths | RS Aggarwal

Mark the tick against the correct answer in the following: The value of $\cot ^{-1}\left(\cot \frac{5 \pi}{4}\right)$ is
A. $\frac{\pi}{4}$
B. $\frac{-\pi}{4}$
C. $\frac{3 \pi}{4}$
D. none of these

Solution:

Option(A) is correct.
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 $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
$\Rightarrow x=\frac{5 \pi}{4} \notin\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
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
$\begin{array}{l} \Rightarrow \cot x=\cot \left(\pi+\frac{\pi}{4}\right)\left(\because \cot \left(\frac{5 \pi}{4}\right)=\cot \left(\pi+\frac{\pi}{4}\right)\right) \\ \Rightarrow \cot x=\cot \left(\frac{\pi}{4}\right)(\because \cot (\pi+\theta)=\cot \theta) \\ \Rightarrow x=\frac{\pi}{4} \end{array}$