# Prosthaphaeresis Formulas/Hyperbolic Cosine minus Hyperbolic Cosine

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## Theorem

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

where $\sinh$ denotes hyperbolic sine and $\cosh$ denotes hyperbolic cosine.

## Proof 1

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

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

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

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

$\blacksquare$

## Proof 2

\(\displaystyle 2 \, \map \sinh {\dfrac {x + y} 2} \, \map \sinh {\dfrac {x - y} 2}\) | \(=\) | \(\displaystyle \dfrac 1 2 \paren {e^{\frac {x + y} 2} - e^{-\frac {x + y} 2} } \paren {e^{\frac {x - y} 2} - e^{-\frac {x - y} 2} }\) | Definition of Hyperbolic Sine | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \paren {e^x - e^y - e^{-y} + e^{-x} }\) | simplifying | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {e^x + e^{-x} } 2 - \dfrac {e^y + e^{-y} } 2\) | rearranging | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \cosh x - \cosh y\) | Definition of Hyperbolic Cosine |

$\blacksquare$

## 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: $8.45$: Sum, Difference and Product of Hyperbolic Functions