Hyperbolic Tangent of Sum

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Theorem

$\map \tanh {a + b} = \dfrac {\tanh a + \tanh b} {1 + \tanh a \tanh b}$

where $\tanh$ denotes the hyperbolic tangent.


Corollary

$\map \tanh {a - b} = \dfrac {\tanh a - \tanh b} {1 - \tanh a \tanh b}$


Proof

\(\ds \map \tanh {a + b}\) \(=\) \(\ds \frac {\map \sinh {a + b} } {\map \cosh {a + b} }\) Definition 2 of Hyperbolic Tangent
\(\ds \) \(=\) \(\ds \frac {\sinh a \cosh b + \cosh a \sinh b} {\cosh a \cosh b + \sinh a \sinh b}\) Hyperbolic Sine of Sum and Hyperbolic Cosine of Sum
\(\ds \) \(=\) \(\ds \frac {\frac {\sinh a} {\cosh a} + \frac {\sinh b} {\cosh b} } {1 + \frac {\sinh a \sinh b} {\cosh a \cosh b} }\) dividing the top and bottom by $\cosh a \cosh b$
\(\ds \) \(=\) \(\ds \frac {\tanh a + \tanh b} {1 + \tanh a \tanh b}\) Definition 2 of Hyperbolic Tangent

$\blacksquare$


Also see


Sources

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