Question

Consider the following statements
Statement I $\tan ^{-1}(\tan x)=x$, if $x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$.
Statement II The value of $\tan ^{-1}\left(\tan \frac{9 \pi}{8}\right)$ is $\frac{\pi}{8}$.
Choose the correct option.

• A. Only I is true
• B. Only II is true
• C. Both I and II are true
• D. Neither I nor II is true

Answer 1

Answer: (C)

We know that, $\tan ^{-1}(\tan x)=x$, if $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
Let $ \theta=\tan ^{-1}\left(\tan \frac{9 \pi}{8}\right) $
$= \tan ^{-1}\left[\tan \left(\pi+\frac{\pi}{8}\right)\right]\left[\because \text { range of } \tan ^{-1} \times \text { is }\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right] $
$=\tan ^{-1}\left[\tan \frac{\pi}{8}\right] [\because \tan (\pi+x)=\tan x]$
$ = \frac{\pi}{8} {\left[\because \frac{\pi}{8} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right] }$