Question

The value of $\cos (2{\cos }^{-1}x+{\sin }^{-1}x)$ at $x=\frac{1}{5}$ is

• A. $\frac{2\sqrt{6}}{5}$
• B. $-\frac{2\sqrt{6}}{5}$
• C. $\frac{3\sqrt{6}}{5}$
• D. None of these

Answer 1

Answer: (B)

$-\frac{2\sqrt{6}}{5}$

Given, $\cos (2{\cos }^{-1}x+{\sin }^{-1}x)$

$=\cos ({\cos }^{-1}x+{\cos }^{-1}x+{\sin }^{-1}x)$

$=\cos (\frac{\pi }{2}+{\cos }^{-1}x)$

$=-\sin \left({\cos }^{-1}x\right)$

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

$=-\sqrt{1-x^2}$

At $x=\frac{1}{5}$

$=-\sqrt{1-\frac{1}{25}}=-\frac{2\sqrt{6}}{5}$