First slide

Properties of ITF

Question

If $\tan (\cos^{- 1} x) = \sin (\cot^{- 1} \frac{1}{2})$ , then $x =$

Easy

A

$\pm \frac{5}{3}$
B

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

$\pm \frac{5}{\sqrt{3}}$
D

None of these

Solution

$\tan (\cos^{- 1} x) = \sin (\cot^{- 1} \frac{1}{2})$
$\begin{array}{l} \Rightarrow \tan (\tan^{- 1} (\frac{\sqrt{1 - x^{2}}}{x})) = \sin (\sin^{- 1} (\frac{2}{\sqrt{5}})) \\ \Rightarrow \frac{\sqrt{1 - x^{2}}}{x} = \frac{2}{\sqrt{5}} \\ \Rightarrow 4 x^{2} = 5 - 5 x^{2} \\ \Rightarrow x = \pm \frac{\sqrt{5}}{3} \end{array}$

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