# Simpson's Formulas/Sine by Sine

## Theorem

$\sin \alpha \sin \beta = \dfrac {\cos \left({\alpha - \beta}\right) - \cos \left({\alpha + \beta}\right)} 2$

where $\sin$ denotes sine and $\cos$ denotes cosine.

## Proof

 $\displaystyle$  $\displaystyle \frac {\cos \left({\alpha - \beta}\right) - \cos \left({\alpha + \beta}\right)} 2$ $\displaystyle$ $=$ $\displaystyle \frac {\left({\cos \alpha \cos \beta + \sin \alpha \sin \beta}\right) - \left({\cos \alpha \cos \beta - \sin \alpha \sin \beta}\right)} 2$ Cosine of Difference and Cosine of Sum $\displaystyle$ $=$ $\displaystyle \frac {2 \sin \alpha \sin \beta} 2$ $\displaystyle$ $=$ $\displaystyle \sin \alpha \sin \beta$

$\blacksquare$