Question

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

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

Answer 1

Answer: (A)

$\cos \left(2{\cos }^{-1}x+{\sin }^{-1}x\right)$
$=\cos \left({\cos }^{-1}x+{\cos }^{-1}x+{\sin }^{-1}x\right)$
$=\cos \left({\cos }^{-1}x+\pi /2\right)=-\sin {\cos }^{-1}x$
$=-\sin {\sin }^{-1}\sqrt{1-x^2}=-\sqrt{1-x^2}$
$=-\sqrt{1-{\left(\frac{1}{5}\right)}^2}=-\sqrt{\frac{24}{25}}=-\frac{2\sqrt{6}}{5}$