^ -1 [x 1-x-x 1-x^2 ]= - Vidyadip

MCQ

$\sin ^{-1}\left[x \sqrt{1-x}-\sqrt{x} \sqrt{1-x^2}\right]=$

A
$\sin ^{-1} x+\sin ^{-1} \sqrt{x}$
✓
$\sin ^{-1} x$$-\sin ^{-1}$$\sqrt{x}$
C
$\sin ^{-1} \sqrt{x}-\sin ^{-1} x$
D
$\sin ^{-1}$$(x-\sqrt{x})$

✓

Answer

Correct option: B.

$\sin ^{-1} x$$-\sin ^{-1}$$\sqrt{x}$

(B) Let $x=\sin \theta$ and $\sqrt{x}=\sin \phi$ Hence
$\therefore \sin ^{-1}\left(x \sqrt{1-x}-\sqrt{x} \sqrt{1-x^2}\right)$
$\begin{array}{l}=\sin ^{-1}\left(\sin \theta \sqrt{1-\sin ^2 \phi}-\sin \phi \sqrt{1-\sin ^2 \theta}\right) \\ =\sin ^{-1}(\sin \theta \cos \phi-\sin \phi \cos \theta) \\ =\sin ^{-1} \sin (\theta-\phi) \\ =\theta-\phi=\sin ^{-1}(x)-\sin ^{-1}(\sqrt{x})\end{array}$

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