Question

The range of the function
$f(x) = \frac{cosec^{-1} x + cos^{-1} (1/x)}{cosec\,x} $ 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

Answer: (B)

$f(x) = \frac{cosec^{-1} x + sec^{-1} x}{cosec\,x} , cos^{-1}(1/x) = sec^{-1} x $
$f(x) = \frac{\pi}{2} sin \,x$
Now, $-1 \le sin\,x \le 1$
$\therefore -\frac{\pi}{2} \le \frac{\pi}{2} \,sin\,x \le \frac{\pi}{2}$
$\Rightarrow - \frac{\pi}{2} \le f(x) \le \frac{\pi}{2}$