# Prosthaphaeresis Formulas/Sine plus Sine

## Theorem

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

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

## Proof

 $\displaystyle$  $\displaystyle 2 \sin \left({\frac {\alpha + \beta} 2}\right) \cos \left({\frac {\alpha - \beta} 2}\right)$ $\displaystyle$ $=$ $\displaystyle 2 \frac {\sin \left({\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2}\right) + \sin \left({\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2}\right)} 2$ Simpson's Formula for Sine by Cosine $\displaystyle$ $=$ $\displaystyle \sin \frac {2 \alpha} 2 + \sin \frac {2 \beta} 2$ $\displaystyle$ $=$ $\displaystyle \sin \alpha + \sin \beta$

$\blacksquare$

## Also reported as

This result is also sometimes reported as:

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

## Linguistic Note

The word prosthaphaeresis or prosthapheiresis is a neologism coined some time in the $16$th century from the two Greek words:

Ian Stewart, in his Taming the Infinite from $2008$, accurately and somewhat diplomatically describes the word as "ungainly".