Here is the proof using extension of well known trigonometrical relations to complex plane - see complex analysis cosh(x+y) = cos(ix+iy) = cos(ix)cos(iy) - sin(ix)sin(iy) = cosh(x)cosh(y) + sinh(x)sinh(y) Solution Here is the proof using extension of well known trigonometrical relations to complex plane - see complex analysis cosh(x+y) = cos(ix+iy) = cos(ix)cos(iy) - sin(ix)sin(iy) = cosh(x)cosh(y) + sinh(x)sinh(y).