The angle addition formulas in “regular” trigonometry state that
$\sin(\alpha+\beta)=\sin\alpha\cos\beta + \cos\alpha\sin\beta$
and
$\cos(\alpha+\beta)=\cos\alpha\cos\beta - \sin\alpha\sin\beta$
It is fairly straightforward and easy to derive these formulas from a figure like this:
It is also easy to prove it using:
$\cos(x) = \frac{e^{ix} + e^{-ix}}2$
and
$\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$
Similar relations hold for the hyperbolic trig functions, namely
$\sinh(\alpha+\beta)=\sinh\alpha\cosh\beta + \cosh\alpha\sinh\beta$
$\cosh(\alpha+\beta)=\cosh\alpha\cosh\beta + \sinh\alpha\sinh\beta$
which again can be computed using
$\cosh(x) = \frac{e^{x} + e^{-x}}2$
$\sinh(x) = \frac{e^{x} - e^{-x}}2$
How would one prove the angle sum formulas for the hyperbolic functions from their geometric definition?