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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $8 \ \text{(i)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.47$: Sum, Difference and Product of Hyperbolic Functions