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

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

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

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

Hyperbolic Sine minus Hyperbolic Sine

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

Hyperbolic Cosine plus Hyperbolic Cosine

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

Hyperbolic Cosine minus Hyperbolic Cosine

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

Also known as

The Prosthaphaeresis Formulas are also known as:

• the factor formulas (or formulae)
• Simpson's formulas (or formulae), although this is usually used for similar results.

Also see

• Simpson's Formulas

Linguistic Note

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

prosthesis, meaning addition

aphaeresis or apheiresis, meaning subtraction.

With the advent of machines to aid the process of arithmetic, this word now has only historical significance.

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

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factor formulae
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factor formulae
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): compound angle formulae (in trigonometry)

• Weisstein, Eric W. "Prosthaphaeresis Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html