prove that sinh(x+y)=sinxcoshy+coshxsinhy

Question

prove the hyperbolic identities

Answer 1

sinh(A)=(eˣ-e⁻ˣ)/2 by definition when A=x, so, if A=x+y:

sinh(x+y)=(e^(x+y)-(e^-(x+y)))/2.

cosh(x)=(eˣ+e⁻ˣ)/2 by definition.

sinh(x)cosh(y)=(e^x-e^-x)(e^y+e^-y)/4=(e^(x+y)+e^(x-y)-e^(y-x)-e^-(x+y))/4.

cosh(x)sinh(y)=(e^x+e^-x)(e^y-e^-y)/4=(e^(x+y)-e^(x-y)+e^(y-x)-e^-(x+y))/4.

When we add these last two equations we get sinh(x)cosh(y)+cosh(x)sinh(y)=

2(e^(x+y)-e^-(x+y))/4=(e^(x+y)-e^-(x+y))/2=sinh(x+y) QED