Hyperbolic Cosine of Sum
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Contents
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
\(\displaystyle \) | \(\) | \(\displaystyle \cosh a \cosh b + \sinh a \sinh b\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac {e^a + e^{-a} } 2 \frac {e^b + e^{-b} } 2 + \frac {e^a - e^{-a} } 2 \frac {e^b - e^{-b} } 2\) | Definition of Hyperbolic Sine and Definition of Hyperbolic Cosine | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac {e^{a + b} + e^{-a + b} + e^{a + b} + e^{-a - b} } 4\) | Exponential of Sum | ||||||||||
\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac {e^{a + b} - e^{-a + b} - e^{a - b} + e^{-a - b} } 4\) | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac {2 e^{a + b} + 2 e^{-\paren {a + b} } } 4\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac {e^{a + b} + e^{-\paren {a + b} } } 2\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \map \cosh {a + b}\) | Definition of Hyperbolic Cosine |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.21$: Addition Formulas
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $5$