728x90
Trigometric and hyperbolic function
$$e^{ix}=cosx+isinx\;\;\;e^{-ix}=cosx-isinx\\cosx=\frac{e^{ix}+e^{-ix}}{2}\;\;\;sinx=\frac{e^{ix}-e^{-ix}}{2i}$$
$$cosz=cosxcoshy-isinxsinhy\\cosz=\frac{e^{iz}+e^{-iz}}{2}=\frac{e^{i(x+yi)}+e^{-i(x+yi)}}{2}\\\Rightarrow \frac{1}{2}(e^{y}(cosx-isinx))+\frac{1}{2}(e^{-y}(cosx+isinx))\\\Rightarrow \frac{1}{2}cosx(e^{y}+e^{-y})+\frac{1}{2i}sinx(e^{y}-e^{-y}$$
$$Similarly\;\;\;sinz=sinxcoshy+icosxsinhy$$
$$\left| cosz \right|^{2} = cos^{2}xcosh^{2}y+sin^{2}xsinh^{2}y\\=cos^{2}x(1+sinh^{2}y)+sin^{2}xsinh^{2}y\\=cos^{2}x+sinh^{2}y
\\Simlarily\;\; \left| sinz \right|^{2}=sin^{2}x+sinh^{2}y$$
$$coshz=\frac{1}{2}(e^{z}+e^{-z})\;\;\;\;\;sinhz=\frac{1}{2}(e^{z}-e^{-z})\\coshiz=cosz\;\;\;\;\;sinhiz=isinz\\cosiz=coshz\;\;\;\;\;siniz=isinhz$$