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

MCQ

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

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

✓

Answer

Correct option: D.

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

We need to find value of $ \sin (\tan^{-1}\text{x})$ Put $\text{y}=\tan^{-1}\text{x}$
$ \Rightarrow \displaystyle \tan {\text{ y} }$
$ \therefore \tan \text{y}=\frac {\sin \text{y}}{\cos \text{y}}$
$\Rightarrow \sin \text{y}=\frac{\tan \text{y}}{\sec \text{y}}$
$ \Rightarrow \sin { \text{y} } =\frac { \text{x} }{ \sqrt { 1+{\text{ x }}^{ 2 } } }$

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