Choose the correct answer from the given four options.If ^ -1 x> … - Vidyadip

MCQ

Choose the correct answer from the given four options.If $\cos^{-1}\text{x}>\sin^{-1}\text{x},$ then:

A
$\frac{1}{\sqrt{2}}<\text{x}\leq1$
B
$0\leq\text{x}<\frac{1}{\sqrt{2}}$
✓
$-1\leq\text{x}<\frac{1}{\sqrt{2}}$
D
$\text{x}>0$

✓

Answer

Correct option: C.

$-1\leq\text{x}<\frac{1}{\sqrt{2}}$

We have, $\cos^{-1}\text{x}>\sin^{-1}\text{x}$
$\Rightarrow\ \frac{\pi}{2}-\sin^{-1}\text{x}>\sin^{-1}\text{x}$
$\Rightarrow\ \frac{\pi}{2}>2\sin^{-1}\text{x}$
$\Rightarrow\ \sin^{-1}\text{x}<\frac{\pi}{4}\ ....(\text{i})$
But $-\frac{\pi}{2}\leq\sin^{-1}\text{x}\leq\frac{\pi}{2}\ ....(\text{ii})$
From $(i)$ and $(ii), -\frac{\pi}{2}\leq\sin^{-1}\text{x}<\frac{\pi}{4}$
$\Rightarrow\ \sin\Big(-\frac{\pi}{2}\Big)\leq\text{x}<\sin\frac{\pi}{4}$
$\Rightarrow\ -1\leq\text{x}<\frac{1}{\sqrt{2}}$

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