Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine

Theorem

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

where $\cosh$ denotes hyperbolic cosine.

$\ds $

$\ds 2 \map \cosh {\frac {x + y} 2} \map \cosh {\frac {x - y} 2}$

$\ds $

$=$

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

$\ds $

$=$

$\ds \cosh \frac {2 x} 2 + \cosh \frac {2 y} 2$

$\ds $

$=$

$\ds \cosh x + \cosh y$

$\blacksquare$

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".

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.44$: Sum, Difference and Product of Hyperbolic Functions