Mark the tick against the correct answer in the following: The value of $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ is A. $\frac{7 \pi}{6}$ B. $\frac{5 \pi}{6}$ C. $\frac{\pi}{6}$ D. none of these

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Mark the tick against the correct answer in the following: The value of $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ is
A. $\frac{7 \pi}{6}$
B. $\frac{5 \pi}{6}$
C. $\frac{\pi}{6}$
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 $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ is
A. $\frac{7 \pi}{6}$
B. $\frac{5 \pi}{6}$
C. $\frac{\pi}{6}$
D. none of these

Solution:

To Find: The value of $\tan ^{-1}\left(\tan \left(\frac{7 \pi}{6}\right)\right)$
Now, let $x=\tan ^{-1}\left(\tan \left(\frac{7 \pi}{6}\right)\right)$
$\Rightarrow \tan x=\tan \left(\frac{7 \pi}{6}\right)$
Here range of principle value of $\tan$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
$\Rightarrow \mathrm{x}=\frac{7 \pi}{6} \notin\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
Hence for all values of $x$ in range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ , the value of $\tan ^{-1}\left(\tan \left(\frac{13 \pi}{6}\right)\right)$ is
$\begin{array}{l} \Rightarrow \tan x=\tan \left(\pi+\frac{\pi}{6}\right)\left(\cdot \tan \left(\frac{7 \pi}{6}\right)=\tan \left(\pi+\frac{\pi}{6}\right)\right) \\ \Rightarrow \tan x=\tan \left(\frac{\pi}{6}\right)(: \tan (\pi+\theta)=\tan \theta) \\ \Rightarrow x=\frac{\pi}{6} \end{array}$