# Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine

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## Contents

## Theorem

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

where $\cosh$ denotes hyperbolic cosine.

## Proof

\(\displaystyle \) | \(\) | \(\displaystyle 2 \map \cosh {\frac {x + y} 2} \map \cosh {\frac {x - y} 2}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \frac {\map \cosh {\dfrac {x + y} 2 + \dfrac {x - y} 2} + \map \cosh {\dfrac {x + y} 2 - \dfrac {x - y} 2} } 2\) | Simpson's Formula for Hyperbolic Cosine by Hyperbolic Cosine | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \cosh \frac {2 x} 2 + \cosh \frac {2 y} 2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \cosh x + \cosh y\) |

$\blacksquare$

## 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

- 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