Question

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

• 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

Answer 1

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

We have $\tan \left({\cos }^{-1}x\right)=\sin \left({\cot }^{-1}\frac{1}{2}\right)$

$\Rightarrow \tan \left[{\tan }^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)\right]=\sin \left({\tan }^{-1}2\right)$

$\Rightarrow \tan \left[{\tan }^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)\right]=\sin \left({\sin }^{-1}\frac{2}{\sqrt{1+4}}\right)$

$\Rightarrow \tan \left[{\tan }^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)\right]=\sin \left({\sin }^{-1}\frac{2}{\sqrt{5}}\right)$

$\Rightarrow \left(\frac{\sqrt{1-x^2}}{x}\right)=\frac{2}{\sqrt{5}}\Rightarrow 5(1-x^2)=4x^2$

$\Rightarrow 9x^2=5\Rightarrow x=\pm \frac{\sqrt{5}}{3}$