Question

The range of $f(x)=\cot ^{-1}(-x)-\tan ^{-1} x+\sec ^{-1} x$ is

• A. $\left[-\frac{\pi}{2}, \frac{3 \pi}{2}\right]$
• B. $\left[\frac{\pi}{2}, \pi\right) \cup\left(\pi, \frac{3 \pi}{2}\right]$
• C. $\left(\frac{-\pi}{2}, \frac{3 \pi}{2}\right)$
• D. $\left(\frac{\pi}{2}, \pi\right) \cup\left(\pi, \frac{3 \pi}{2}\right)$

Answer 1

Answer: (B)

We have $f ( x )=\pi-\cot ^{-1} x -\tan ^{-1} x +\sec ^{-1} x =\pi-\frac{\pi}{2}+\sec ^{-1} x =\frac{\pi}{2}+\sec ^{-1} x$
As domain of $f(x)$ is $(-\infty,-1] \cup[1, \infty)$ $\left(\right.$ As $\left.\cot ^{-1}(-x)=\pi-\cot ^{-1} x\right)$
$\therefore$ Range of $f ( x )$ is $\left[\frac{\pi}{2}, \pi\right) \cup\left(\pi, \frac{3 \pi}{2}\right]$.