Hyperbolic Sine of Sum

Theorem

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

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

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

$\ds \sinh a \cosh b + \cosh a \sinh b$

$=$

$\ds \frac {e^a - e^{-a} } 2 \frac {e^b + e^{-b} } 2 + \frac {e^a + e^{-a} } 2 \frac {e^b - e^{-b} } 2$

Definition of Hyperbolic Sine and Definition of Hyperbolic Cosine

$\ds $

$=$

$\ds \frac {e^{a + b} - e^{-a + b} + e^{a - b} - e^{-a - b} } 4$

$\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 $

$=$

$\ds \map \sinh {a + b}$

Definition of Hyperbolic Sine

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.20$: Addition Formulas
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hyperbolic function