Q. The range of the function $f ( x )=\sec ^{-1} x +\tan ^{-1} x +\operatorname{cosec}^{-1} x$, is
Inverse Trigonometric Functions
Solution:
The domain of the function $f ( x )$ is $(-\infty,-1] \cup[1, \infty)$
$f ( x )=\frac{\pi}{2}+\tan ^{-1} x $
$\tan ^{-1} x \in\left(\frac{-\pi}{2}, \frac{-\pi}{4}\right] \cup\left[\frac{\pi}{4}, \frac{\pi}{2}\right)$
Hence, range of $f(x)$ is $\left(0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \pi\right)$.
