# Prosthaphaeresis Formulas/Sine plus Sine

## Theorem

$\sin \alpha + \sin \beta = 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$

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

## Proof

 $\displaystyle$  $\displaystyle 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$ $\displaystyle$ $=$ $\displaystyle 2 \frac {\map \sin {\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2} + \map \sin {\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2} } 2$ 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:

$\dfrac {\sin \alpha + \sin \beta} 2 = \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$

## 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".