# How do you simplify Sin[cos^-1(5/13) - cos^-1(4/5)]?

May 26, 2016

$\pm \frac{33}{65} , \pm \frac{63}{65}$.

#### Explanation:

Let a = cos^(-1)(5/13). Then $\cos a = \frac{5}{13} > 0.$ a is in the 1st

quadrant or in the 4th. So, sin a =+-12/13.

Let $b = {\cos}^{- 1} \left(\frac{4}{5}\right)$. Then $\cos b = \frac{4}{5} > 0$. b is in the 1st

quadrant or in the 4th. So, $\sin b = \pm \frac{3}{5}$.

Now, the given expression is

$\sin \left(a - b\right) = \sin a \cos b - \cos a \sin b$

$= \left(\pm \frac{12}{13}\right) \left(\frac{4}{5}\right) - \left(\frac{5}{13}\right) \left(\pm \frac{3}{5}\right)$

$= \pm \frac{48}{65}$ - or +$\frac{3}{13}$

$= \pm \frac{63}{65} , \pm \frac{33}{65}$.