(1)

\begin{align*} \cos z\pm i\sin z & =\frac{e^{iz}+e^{-iz}}{2}\pm i\frac{e^{iz}-e^{-iz}}{2i}\\ & =\frac{1}{2}\left(e^{iz}+e^{-iz}\pm\left(e^{iz}-e^{-iz}\right)\right)\\ & =\frac{1\pm1}{2}e^{iz}+\frac{1\mp1}{2}e^{-iz}\\ & =e^{\pm iz} \end{align*}

(2)

\begin{align*} \sin z\pm i\cos z & =\pm i\left(\cos z\mp i\sin z\right)\\ & =\pm ie^{\mp iz} \end{align*}

(3)

\begin{align*} \cosh z\pm\sinh z & =\cos\left(iz\right)\pm i^{-1}\sin\left(iz\right)\\ & =\cos\left(iz\right)\mp i\sin\left(iz\right)\\ & =e^{\mp i\left(iz\right)}\\ & =e^{\pm z} \end{align*}

(4)

\begin{align*} \sinh z\pm\cosh z & =\pm\left(\cosh z\pm\sinh z\right)\\ & =\pm e^{\pm z} \end{align*}