First slide

Properties of ITF

Question

Solution of the equation $\tan (\cos^{- 1} x) = \sin (\cot^{- 1} \frac{1}{2})$ is

Moderate

A

$x = \pm \frac{\sqrt{7}}{3}$
B

$x = \pm \frac{\sqrt{5}}{3}$
C

$x = \pm \frac{3 \sqrt{5}}{2}$
D

none of these

Solution

We have, $\tan (\cos^{- 1} x) = \sin (\cot^{- 1} \frac{1}{2})$
$\begin{array}{l} \Rightarrow \tan [\tan^{- 1} (\frac{\sqrt{1 - x^{2}}}{x})] = \sin (\tan^{- 1} 2) \\ \Rightarrow \tan [\tan^{- 1} (\frac{\sqrt{1 - x^{2}}}{x})] = \sin (\sin^{- 1} \frac{2}{\sqrt{1 + 4}}) \\ \Rightarrow \tan [\tan^{- 1} (\frac{\sqrt{1 - x^{2}}}{x})] = \sin (\sin^{- 1} \frac{2}{\sqrt{5}}) \\ \Rightarrow \frac{\sqrt{1 - x^{2}}}{x} = \frac{2}{\sqrt{5}} \Rightarrow 5 (1 - x^{2}) = 4 x^{2} \\ \Rightarrow 9 x^{2} = 5, ∴ x = \pm (\frac{\sqrt{5}}{3}) . \end{array}$

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