Question

If $\tan \left({\cos }^{-1}x\right)=\sin \left({\cot }^{-1}\frac{1}{2}\right)$, then find the value of $x$

• A. $x=-\frac{\sqrt{5}}{3}$
• B. $x=+\frac{\sqrt{5}}{3}$
• C. $x=-\frac{2}{3}$
• D. $x=+\frac{2}{3}$

Answer 1

The correct option is B $x=+\frac{\sqrt{5}}{3}$

Given, $\tan \left({\cos }^{-1}x\right)=\sin \left({\cot }^{-1}\frac{1}{2}\right)$
$\Rightarrow \tan \left({\tan }^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)\right)=\sin \left({\sin }^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)$
$\Rightarrow \frac{\sqrt{1-x^2}}{x}=\frac{2}{\sqrt{5}}\Rightarrow x=\frac{\sqrt{5}}{3}$