Question

If $\cos^{-1} x > \sin^{-1} x ,$ then $ x $ lies in the interval

• A. $\left(\frac{1}{2},1\right]$
• B. ( 0, 1 ]
• C. $\left[-1\frac{1}{\sqrt{2}}\right)$
• D. [ - 1, 1 ]
• E. [ 0, 1 ]

Answer 1

Answer: (C)

We know that, $\cos ^{-1} X$ and $\sin ^{-1} x$ exist for $X \in[-1,1]$
Now, $\cos ^{-1} X>\sin ^{-1} X$
$\Rightarrow \cos ^{-1} x>\frac{\pi}{2}-\cos ^{-1} x$
$\Rightarrow 2 \cos ^{-1} x>\frac{\pi}{2}$
$\Rightarrow \cos ^{-1} x>\frac{\pi}{4}$
$\Rightarrow x \in\left[-1, \frac{1}{\sqrt{2}}\right)$