^ -1 1-x^2 x equal to : - Vidyadip

MCQ

$\cot ^{-1} \frac{\sqrt{1-x^2}}{x}$ equal to :

A
$\sin ^{-1}\left(\frac{1}{x}\right)$
✓
$\operatorname{cosec}^{-1}\left(\frac{1}{x}\right)$
C
$\tan ^{-1}\left(\frac{1}{x}\right)$
D
None of these

✓

Answer

Correct option: B.

$\operatorname{cosec}^{-1}\left(\frac{1}{x}\right)$

(B)
$\cot ^{-1} \frac{\sqrt{1-x^2}}{x}$
taking $x=\sin \theta$
$\Rightarrow \quad \cot ^{-1} \frac{\sqrt{1-\sin ^2 \theta}}{\sin \theta}=\cot ^{-1} \frac{\cos \theta}{\sin \theta}$
$\Rightarrow \quad \cot ^{-1}(\cot \theta)=\theta=\sin ^{-1} x=\operatorname{cosec}^{-1}\left(\frac{1}{x}\right)$
Hence correct option is (B).

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