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Trigometric and hyperbolic function
eix=cosx+isinxe−ix=cosx−isinxcosx=eix+e−ix2sinx=eix−e−ix2i
cosz=cosxcoshy−isinxsinhycosz=eiz+e−iz2=ei(x+yi)+e−i(x+yi)2⇒12(ey(cosx−isinx))+12(e−y(cosx+isinx))⇒12cosx(ey+e−y)+12isinx(ey−e−y
Similarlysinz=sinxcoshy+icosxsinhy
|cosz|2=cos2xcosh2y+sin2xsinh2y=cos2x(1+sinh2y)+sin2xsinh2y=cos2x+sinh2ySimlarily|sinz|2=sin2x+sinh2y
coshz=12(ez+e−z)sinhz=12(ez−e−z)coshiz=coszsinhiz=isinzcosiz=coshzsiniz=isinhz