First slide

Properties of ITF

Question

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

Moderate

A

$\sqrt{\frac{x^{2} + 2}{x^{2} + 3}}$
B

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

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

none of these

Solution

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

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