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

Question

$\sin (\tan^{–1} x), | x | < 1$ is equal to

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Answer

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

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