Prosthaphaeresis Formulas/Hyperbolic Cosine minus Hyperbolic Cosine/Proof 1

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

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

$\blacksquare$