Prosthaphaeresis Formulas

Theorem

Sine plus Sine

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

Sine minus Sine

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

Cosine plus Cosine

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

Cosine minus Cosine

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

Hyperbolic Sine plus Hyperbolic Sine

$\sinh x + \sinh y = 2 \sinh \left({\dfrac {x + y} 2}\right) \cosh \left({\dfrac {x - y} 2}\right)$

Hyperbolic Sine minus Hyperbolic Sine

$\sinh x - \sinh y = 2 \cosh \left({\dfrac {x + y} 2}\right) \sinh \left({\dfrac {x - y} 2}\right)$

Hyperbolic Cosine plus Hyperbolic Cosine

$\cosh x + \cosh y = 2 \cosh \left({\dfrac {x + y} 2}\right) \cosh \left({\dfrac {x - y} 2}\right)$

Hyperbolic Cosine minus Hyperbolic Cosine

$\cosh x - \cosh y = 2 \, \map \sinh {\dfrac {x + y} 2} \, \map \sinh {\dfrac {x - y} 2}$

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".