Solve ( ^ -1 x ),|x|<1 is equal to - Vidyadip

MCQ

Solve $\sin \left(\tan ^{-1} x\right),|x|<1$ is equal to

A
$\frac{1}{\sqrt{1+x^{2}}}$
B
$\frac{1}{\sqrt{1-x^{2}}}$
✓
$\frac{x}{\sqrt{1+x^{2}}}$
D
$\frac{x}{\sqrt{1-x^{2}}}$

✓

Answer

Correct option: C.

$\frac{x}{\sqrt{1+x^{2}}}$

c
$\tan y=x \Rightarrow \sin y=\frac{x}{\sqrt{1+x^{2}}}$

Let $\tan ^{-1} x=y .$ Then

$y=\sin ^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right)$ $\Rightarrow \tan ^{-1} x=\sin ^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right)$

$\Rightarrow \sin \left(\tan ^{-1} x\right)=\sin \left(\sin ^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right)\right)=\frac{x}{\sqrt{1+x^{2}}}$

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