Hyperbolic Cosine of Sum

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Theorem

$\map \cosh {a + b} = \cosh a \cosh b + \sinh a \sinh b$

where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine.


Corollary

$\map \cosh {a - b} = \cosh a \cosh b - \sinh a \sinh b$


Proof

\(\ds \) \(\) \(\ds \cosh a \cosh b + \sinh a \sinh b\)
\(\ds \) \(=\) \(\ds \frac {e^a + e^{-a} } 2 \frac {e^b + e^{-b} } 2 + \frac {e^a - e^{-a} } 2 \frac {e^b - e^{-b} } 2\) Definition of Hyperbolic Sine and Definition of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \frac {e^{a + b} + e^{-a + b} + e^{a + b} + e^{-a - b} } 4\) Exponential of Sum
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \frac {e^{a + b} - e^{-a + b} - e^{a - b} + e^{-a - b} } 4\)
\(\ds \) \(=\) \(\ds \frac {2 e^{a + b} + 2 e^{-\paren {a + b} } } 4\)
\(\ds \) \(=\) \(\ds \frac {e^{a + b} + e^{-\paren {a + b} } } 2\)
\(\ds \) \(=\) \(\ds \map \cosh {a + b}\) Definition of Hyperbolic Cosine

$\blacksquare$


Also see


Sources

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