Question

The value of $x$ satisfying $\tan \left({\sec }^{-1}x\right)=\sin \left(co{s}^{-1}\frac{1}{\sqrt{5}}\right)$ is

• A. $\pm \frac{3}{\sqrt{5}}$
• B. $\pm \frac{5}{\sqrt{3}}$
• C. $\pm \frac{\sqrt{2}}{3}$
• D. $\pm \frac{3}{5}$

Answer 1

Answer: (A)

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

We have;

$\tan \left({\sec }^{-1}x\right)=\sin {\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

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

$⟹\sqrt{x^2-1}=\frac{2}{\sqrt{5}}$

Squaring on both sides; we get

$⟹x^2-1=\frac{4}{5}$

$⟹x^2=\frac{4}{5}+1=\frac{9}{5}$

$⟹x=\pm \frac{3}{\sqrt{5}}$

Hence;

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