Question

The range of the the function, $f\left(x\right)=({\cot }^{-1}x+{\sec }^{-1}x+cose{c}^{-1}x)$ is

• A. $(\frac{\pi }{2},\frac{3\pi }{2})$
• B. $(\frac{\pi }{2},\frac{3\pi }{4}]\cup [\frac{5\pi }{4},\frac{3\pi }{2})$
• C. $[\frac{\pi }{2},\pi )\cup (\pi ,\frac{3\pi }{2}]$
• D. $(\frac{\pi }{2},\pi )\cup (\pi ,\frac{3\pi }{2})$

Answer 1

The correct option is B $(\frac{\pi }{2},\frac{3\pi }{4}]\cup [\frac{5\pi }{4},\frac{3\pi }{2})$

Given,

$f\left(x\right)=({\cot }^{-1}x+{\sec }^{-1}x+cose{c}^{-1}x)$

Now,

$0<co{t}^{-1}x<\pi$ ..(1)
$0\le {sec}^{-1}x<\frac{\pi }{2}$

and

$\frac{\pi }{2}<{sec}^{-1}x\le \pi$ ...(2)

$-\frac{\pi }{2}\le {cosec}^{-1}x<0$ and $0<{cosec}^{-1}\le \frac{\pi }{2}$ ..(3)

Adding (1) , (2) and (3), we get
$\frac{\pi }{2}<co{t}^{-1}x+{sec}^{-1}x+{cosec}^{-1}x\le \frac{3\pi }{4}$ and $\frac{5\pi }{4}\le co{t}^{-1}x+{sec}^{-1}x+{cosec}^{-1}x<\frac{3\pi }{2}$