# How do you simplify Cos(sin^-1 u + cos^-1 v)?

Apr 13, 2016

$v \cdot \sqrt{1 - {u}^{2}} - u \cdot \sqrt{1 - {v}^{2}}$

#### Explanation:

$\cos \left({\sin}^{- 1} u + {\cos}^{- 1} v\right) =$

Making
${\sin}^{- 1} u = \alpha$ => $\sin \alpha = u$
${\cos}^{- 1} v = \beta$ => $\cos \beta = v$

And using the formula

$\cos \left(\alpha + \beta\right) = \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta$

We get

$\cos \left(\alpha + \beta\right) = \sqrt{1 - {\sin}^{2} \alpha} \cdot \cos \beta - \sin \alpha \cdot \sqrt{1 - {\cos}^{2} \beta}$
$= \sqrt{1 - {u}^{2}} \cdot v - u \cdot \sqrt{1 - {v}^{2}}$
$= v \cdot \sqrt{1 - {u}^{2}} - u \cdot \sqrt{1 - {v}^{2}}$