Question

The value of $\cos [2{\cos }^{-1}\frac{1}{5}+{\sin }^{-1}\frac{1}{5}]$ is

• A. $\frac{2\sqrt{6}}{5}$
• B. $-\frac{2\sqrt{6}}{5}$
• C. $\frac{1}{5}$
• D. $-\frac{1}{5}$

Answer 1

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

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