Question

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

• A. $-\sqrt{\frac{3}{5}}$
• B. $2\sqrt{\frac{9}{5}}$
• C. $-2\frac{\sqrt{6}}{8}$
• D. $-2\frac{\sqrt{6}}{5}$
  
  

Answer 1

The correct option is D

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

We have,

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

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

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

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

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

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

Since, $x=\frac{1}{5}$

Therefore,

$\Rightarrow -\sqrt{1-(1/5{)}^2}$

$\Rightarrow -\sqrt{\left(24/25\right)}$

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

Hence, this is the answer.