For \(z\in\CC\), the hyperbolic cosine and hyperbolic sine are \[ \begin{aligned} \cosh(z)&\coloneqq\frac{e^{z}+e^{-z}}{2},\\ \sinh(z)&\coloneqq\frac{e^{z}-e^{-z}}{2}. \end{aligned} \]
With \(iz\) in place of \(z\), the same expressions give the trigonometric functions: \(\cos(z)=\frac{e^{iz}+e^{-iz}}{2}=\cosh(iz)\) and \(\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}=-i\sinh(iz)\).
How are $\cosh$ and $\sinh$ written as exponentials?