Question

Evaluate: $\cos ({\sin }^{-1}x+2{\cos }^{-1}x)$ at $x=\frac{1}{5}$ is

• A. $\frac{-2\sqrt{6}}{5}$
• B. $\frac{-\sqrt{6}}{5}$
• C. $\frac{2\sqrt{6}}{5}$
• D. $\frac{\sqrt{6}}{5}$

Answer 1

The correct option is A $\frac{-2\sqrt{6}}{5}$

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

$=\cos (\frac{\pi }{2}+{\cos }^{-1}x)=-\sin \left({\cos }^{-1}x\right)=-\sin \left({\sin }^{-1}\sqrt{1-x^2}\right)=-\sqrt{1-x^2}$ (Since x is positive)

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