Hyperbolic Cosine minus Hyperbolic Cosine/Proof 1
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Theorem
- $\cosh x - \cosh y = 2 \map \sinh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}$
Proof
| \(\ds \) | \(\) | \(\ds 2 \, \map \sinh {\frac {x + y} 2} \, \map \sinh {\frac {x - y} 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \, \frac {\map \cosh {\dfrac {x + y} 2 + \dfrac {x - y} 2} - \map \cosh {\dfrac {x + y} 2 - \dfrac {x - y} 2} } 2\) | Werner Formula for Hyperbolic Sine by Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cosh \frac {2 x} 2 - \cosh \frac {2 y} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cosh x - \cosh y\) |
$\blacksquare$