Question

The range of the function, $f\left(x\right)={\cot }^{-1}x+{\sec }^{-1}x+{\text{cosec}}^{-1}x$ 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,
$f\left(x\right)={\cot }^{-1}x+{\sec }^{-1}x+{\text{cosec}}^{-1}x$
$\because {\sec }^{-1}x+{\text{cosec}}^{-1}x=\frac{\pi }{2}$
$f\left(x\right)={\cot }^{-1}x+\frac{\pi }{2}$
Now, domain of $f\left(x\right)$ is
$x\in (-\infty ,-1]\cup [1,\infty )$

For the above domain
${\cot }^{-1}x\in (0,\frac{\pi }{4}]\cup [\frac{3\pi }{4},\pi )$
$\Rightarrow {\cot }^{-1}x+\frac{\pi }{2}\in (\frac{\pi }{2},\frac{3\pi }{4}]\cup [\frac{5\pi }{4},\frac{3\pi }{2})$