Question

Consider a function $f$ defined by $f(x)=\sin ^{-1}\left(\frac{2 \tan \frac{x}{2}}{1+\tan ^2 \frac{x}{2}}\right)$ then

• A. Range of $f(x)$ is $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$
• B. $y=f(x)$ and $y=\sin ^{-1}(\sin x)$ have same graphs.
• C. $y = f ( x )$ is periodic with period $2 \pi$.
• D. Number of solutions of $f ( x )=0$ in $[0,4 \pi]$ is 5 .

Answer 1

Answer: (A), (C)

$f ( x )=\sin ^{-1}(\sin x ), x \neq(2 n +1) \pi$
$\Rightarrow f(x)$ and $y=\sin ^{-1}(\sin x)$ are not identical

Range of $f ( x )=\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ and period $=2 \pi$