Hyperbolic functions/R/Elementary properties/Fact

The functions hyperbolic sine and hyperbolic cosine have the following properties.

1. ${}\cosh x+\sinh x=e^{x}\,.$

2. ${}\cosh x-\sinh x=e^{-x}\,.$

3. ${}(\cosh x)^{2}-(\sinh x)^{2}=1\,.$

Write a proof