Question

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

• A. $\frac{-\sqrt{24}}{5}$
• B. $\frac{\sqrt{16}}{5}$
• C. $\frac{\sqrt{24}}{5}$
• D. $\frac{-\sqrt{16}}{5}$

Answer 1

The correct option is A $\frac{-\sqrt{24}}{5}$

Let $y=\cos [2{\cos }^{-1}x+{\sin }^{-1}x]$
As ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$ for $|x|\le 1,$
$y=\cos [\frac{\pi }{2}+{\cos }^{-1}x]=-\sin \left({\cos }^{-1}x\right)$

Put $x=\frac{1}{5}$
$y=-\sin \left({\cos }^{-1}\frac{1}{5}\right)$
$\Rightarrow y=-\sin \left({\sin }^{-1}\sqrt{1-{\left(\frac{1}{5}\right)}^2}\right)$
$\Rightarrow y=-\sin \left({\sin }^{-1}\frac{\sqrt{24}}{5}\right)=-\frac{\sqrt{24}}{5}$