Question

$\cos(2 \cos^{-1} \frac {1}{5}+\sin^{-1} \frac {1}{5})$ is euqal to

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

Answer 1

Answer: (B)

$\cos \left\{2 \cos ^{-1} \frac{1}{5}+\sin ^{-1} \frac{1}{5}\right\}$
$=\cos \left\{\cos ^{-1} \frac{1}{5}+\left(\cos ^{-1} \frac{1}{5}+\sin ^{-1} \frac{1}{5}\right)\right\}$
$=\cos \left\{\cos ^{-1} \frac{1}{5}+\frac{\pi}{2}\right\}$
$\left(\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}, \forall x \in R\right)$

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