First slide

Properties of ITF

Question

The value of $\sin [\cot^{- 1} \{\cos (\tan^{- 1} x)\}],$ is

Moderate

A

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

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

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

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

Solution

We have,
$\sin [\cot^{- 1} \{\cos (\tan^{- 1} x)\}]$
$= \sin [\cot^{- 1} \{\frac{1}{\sqrt{1 + \tan^{2} (\tan^{- 1} x)}}\}]$
$= \sin \{\cot^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})\}$
$= \frac{1}{\sqrt{1 + \cot^{2} \{\cot^{- 1} \frac{1}{\sqrt{1 + x^{2}}}\}}} = \frac{1}{\sqrt{1 + \frac{1}{1 + x^{2}}}}$
$= \sqrt{\frac{1 + x^{2}}{2 + x^{2}}}$

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