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 sinh

^{2}(x).
For example, suppose we start with

cos^{2}(x) + sin^{2}(x) = 1.

The hyperbolic equivalent is therefore

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

A more complicated identity is

sin(3x) = 3cos^{2}(x)sin(x) - sin^{3}(x).

This becomes

sinh(3x) = 3cosh^{2}(x)sinh(x) + sinh^{3}(x).