Werner Formulas/Hyperbolic Cosine by Hyperbolic Cosine

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Theorem

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

where $\cosh$ denotes hyperbolic cosine.


Proof

\(\ds \) \(\) \(\ds \frac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2\)
\(\ds \) \(=\) \(\ds \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \cosh \paren {x - y} } 2\) Hyperbolic Cosine of Sum
\(\ds \) \(=\) \(\ds \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \paren {\cosh x \cosh y - \sinh x \sinh y} } 2\) Hyperbolic Cosine of Difference
\(\ds \) \(=\) \(\ds \frac {2 \cosh x \cosh y} 2\)
\(\ds \) \(=\) \(\ds \cosh x \cosh y\)

$\blacksquare$


Also presented as

This result can also be seen presented as:

$2 \cosh x \cosh y = \cosh \paren {x + y} + \cosh \paren {x - y}$


Sources

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