Answer
$\frac{4\sqrt 6}{25}$
Work Step by Step
$\sin$(2$\cos^{-1}\frac{1}{5})$
First let $\theta$ = $\cos^{-1}$ so $cos\theta$ = $\frac{1}{5}$
Then use the Pythagorean Theorem:
$A^{2}+B^{2}=C^{2}$
$\sqrt 5^{2}-\sqrt 1^{2}$ = $\sqrt 24 = 2\sqrt 6$
Find the new sin$\theta$:
Sin$\theta$ = $\frac{Opposite}{Hypotenuse}$
So: Sin$\theta$ = $\frac{2\sqrt 6}{5}$
Substitue cos in the original problem with sin$\theta$ and solve:
$2(\frac{2\sqrt 6}{5})(\frac{1}{5})$ = $\frac{4\sqrt 6}{25}$
