Question

If $\theta ={\sin }^{-1}x+{\cos }^{-1}x-{\tan }^{-1}x$, $x\ge 0$, then the smallest interval in which $\theta$ lies is-

• A. $\frac{\pi }{2}\le \theta \le \frac{3\pi }{4}$
• B. $0\le \theta \le \frac{\pi }{4}$
• C. $-\frac{\pi }{4}\le \theta \le 0$
• D. $\frac{\pi }{4}\le \theta \le \frac{\pi }{2}$

Answer 1

The correct option is D $\frac{\pi }{4}\le \theta \le \frac{\pi }{2}$

Given, $\theta ={\sin }^{-1}x+{\cos }^{-1}x-{\tan }^{-1}xx\ge 0$
$=\frac{\pi }{2}-{\tan }^{-1}x,1\ge x\ge 0$
Therefore the smallest interval in which $\theta$ lies is $[\frac{\pi }{4},\frac{\pi }{2}]$