双曲線関数
sinh  ( α + β ) = sinh  α cosh  β + cosh  α sinh  β sinh  ( α − β ) = sinh  α cosh  β − cosh  α sinh  β cosh  ( α + β ) = cosh  α cosh  β + sinh  α sinh  β cosh  ( α − β ) = cosh  α cosh  β − sinh  α sinh  β tanh  ( α + β ) = tanh  α + tanh  β 1 + tanh  α tanh  β tanh  ( α − β ) = tanh  α − tanh  β 1 − tanh  α tanh  β {\displaystyle {\begin{aligned}\sinh(\alpha +\beta )&=\sinh \alpha \cosh \beta +\cosh \alpha \sinh \beta \\\sinh(\alpha -\beta )&=\sinh \alpha \cosh \beta -\cosh \alpha \sinh \beta \\\cosh(\alpha +\beta )&=\cosh \alpha \cosh \beta +\sinh \alpha \sinh \beta \\\cosh(\alpha -\beta )&=\cosh \alpha \cosh \beta -\sinh \alpha \sinh \beta \\\tanh(\alpha +\beta )&={\frac {\tanh \alpha +\tanh \beta }{1+\tanh \alpha \tanh \beta }}\\\tanh(\alpha -\beta )&={\frac {\tanh \alpha -\tanh \beta }{1-\tanh \alpha \tanh \beta }}\end{aligned}}}