We will show that for any real element x, y the trigonometric formula sinh(x + y) = sinh(x)cosh(y) +cosh(x)sinh(y)

Difficult Proof/Demonstration

We start from the left hand side of the equality:

sinh(x+y)=ex+ye(x+y)2=ex+yexy2=2ex+y2exy4=2ex+y2exy+(exyex+y)(exyex+y)4=2ex+y+(exyex+y)(exyex+y)2exy4=ex+y+ex+y+(exyex+y)(exyex+y)exyexy4=ex+yexy+(exyex+y)+ex+y(exyex+y)exy4=(ex+y+exyex+yexy4)+(ex+yexy+ex+yexy4)=(exex2)(ey+ey2)+(ex+ex2)(eyey2)=sinhxcoshy+coshxsinhy

Easy Proof/Demonstration

We start from the right hand side of the equality:

sinhxcoshy+coshxsinhy=(exex2)(ey+ey2)+(ex+ex2)(eyey2)=(ex+y+exyex+yexy4)+(ex+yexy+ex+yexy4)=2ex+y2exy4=ex+yexy2=ex+ye(x+y)2=sinh(x+y)