# Hyperbolic functions/R/Elementary properties/Fact

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

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

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

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

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