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