Question

The range of the function $f\left(x\right)=\frac{cose{c}^{-1}x+{\cos }^{-1}\left(\frac{1}{x}\right)}{cosecx}$ is

• A. $(0,\pi )$
• B. $[-\frac{\pi }{2},\frac{\pi }{2}]$
• C. $[-\frac{\pi }{2},0]$
• D. $(-\frac{\pi }{2},\frac{\pi }{2})$

Answer 1

The correct option is B $[-\frac{\pi }{2},\frac{\pi }{2}]$

Given $f\left(x\right)=\frac{cose{c}^{-1}x+{\cos }^{-1}\left(1/x\right)}{cosecx}$

$f\left(x\right)=\frac{cose{c}^{-1}x+{\sec }^{-1}x}{co\sec x},(\because co{s}^{-1}\left(1/x\right)={\sec }^{-1}x)$
$f\left(x\right)=\frac{\pi }{2}\sin x$
Now $-1\le \sin x\le 1$
$\therefore -\frac{\pi }{2}\le \frac{\pi }{2}\sin x\le \frac{\pi }{2}$
$\Rightarrow -\frac{\pi }{2}\le f\left(x\right)\le \frac{\pi }{2}$