First slide

Properties of ITF

Question

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

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

Let $\cot^{- 1} x = θ \Rightarrow \cot θ = x$
$∴ \sin (\cot^{- 1} x) = \sin θ = \frac{1}{cosec θ} = \frac{1}{\sqrt{1 + \cot^{2} θ}} = \frac{1}{\sqrt{1 + x^{2}}}$
Thus, $\cos [\tan^{- 1} [\sin (\cot^{- 1} x)]]$
$= \cos [\tan^{- 1} \frac{1}{\sqrt{1 + x^{2}}}] = \cos φ$
$(Put \tan^{- 1} \{\frac{1}{\sqrt{1 + x^{2}}}\} = ϕ \Rightarrow \tan ϕ = \frac{1}{\sqrt{1 + x^{2}}})$
$= \frac{1}{\sec ϕ} = \frac{1}{\sqrt{1 + \tan^{2} ϕ}} = \frac{1}{\sqrt{1 + \frac{1}{1 + x^{2}}}} = \sqrt{\frac{1 + x^{2}}{2 + x^{2}}}$

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