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