& \sinh (x \pm y) \equiv \sinh x \cosh y \pm \cosh x \sinh y \\ & \cosh (x \pm y) \equiv \cosh x \cosh y \pm \sinh x \sinh y \\ & \tanh (x \pm y) \equiv \frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh ...& \sinh (x \pm y) \equiv \sinh x \cosh y \pm \cosh x \sinh y \\ & \cosh (x \pm y) \equiv \cosh x \cosh y \pm \sinh x \sinh y \\ & \tanh (x \pm y) \equiv \frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y} \tanh \frac{x}{2} & \equiv \frac{\sinh x}{\cosh x+1} \equiv \frac{\cosh x-1}{\sinh x} & \cosh x \cosh y \equiv \frac{1}{2}[\cosh (x+y)+\cosh (x-y)] \\ \[\begin{array}{l}\sinh x \equiv-i \sin (i x) \\ \cosh x \equiv \cos (i x) \\ \tanh x \equiv-i \tan (i x)\end{array} \notag\]
