Solution
= Solution
Substitute the exponential definitions of <hyperbolic cosine> and <hyperbolic sine>. The cross terms cancel:
$$
\begin{aligned}
\cosh x\cosh y-\sinh x\sinh y
&=\frac{(e^x+e^{-x})(e^y+e^{-y})-(e^x-e^{-x})(e^y-e^{-y})}{4}\\
&=\frac{e^{x-y}+e^{-x+y}}2=\cosh(x-y).
\end{aligned}
$$
This proves the identity for real arguments, and the same algebra proves it for complex arguments.