# If $tan \left ( cos^{-1}x \right )= Sin\left ( cot^{-1}\frac{1}{2} \right )$   then x is equal to Option 1) $\pm \frac{5}{3}$ Option 2) $\pm \frac{\sqrt{5}}{3}$ Option 3) $\pm \frac{5}{\sqrt{3}}$ Option 4) None of these

D Divya Saini

As we learnt in

Relation between all the Inverse Trigonometric Functions -

$\sin ^{-1}x = \tan ^{-1}\frac{x}{\sqrt{1-x^{2}}} = cosec ^{-1}\frac{1}{x} = \cos ^{-1}\sqrt{1-x^{2}} = \cot ^{-1}\frac{\sqrt{1-x^{2}}}{x} = \sec ^{-1}\frac{1}{\sqrt{1-x^{2}}}$

- wherein

This happens with domain of functions, x$>$0

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

$\frac{\sqrt{1-x^{2}}}{x}\:=\:\frac{2}{\sqrt{5}}$

$\therefore 4x^{2}=5-5x^{2}$

$\Rightarrow x^{2}=\frac{5}{3} \Rightarrow x=\:\frac{\pm \sqrt{5}}{3}$

Option 1)

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

Incorrect

Option 2)

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

Correct

Option 3)

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

Incorrect

Option 4)

None of these

Incorrect

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