Osborne's rule

This is a mathematical tool for transforming trigonometric identities into corresponding hyperbolic ones.

Start with a standard identity, and multiply out fully in terms of sines and cosines. Then replace the functions with their hyperbolic analogues. Finally, reverse the sign of any term involving sinh²(x). For example, suppose we start with

cos2(x) + sin2(x) = 1.

The hyperbolic equivalent is therefore

cosh2(x) - sinh2(x) = 1.

A more complicated identity is

sin(3x) = 3cos2(x)sin(x) - sin3(x).

This becomes

sinh(3x) = 3cosh2(x)sinh(x) + sinh3(x).