# How do you simplify sin(cos^-1(2/3)-sin^-1(2/3))?

Jul 29, 2016

$\frac{1}{9}$

#### Explanation:

Use ${\sin}^{- 1} x + {\cos}^{- 1} x = \frac{\pi}{2}$. Here, $x = \frac{2}{3}$.

Let ${\cos}^{- 1} \left(\frac{2}{3}\right) = a$. Then $\cos a = \frac{2}{3}$.

Now, the given expression is

$\sin \left(2 a - \frac{\pi}{2}\right)$

$= - \cos 2 a$

$= 1 - 2 {\cos}^{2} a$

$= \left(1 - \frac{8}{9}\right)$

$= \frac{1}{9}$