Products and powers of trigonometric terms involving sin, cos, sinh and cosh are combined into a sum of trigonometric terms by repeated application of the transformations

where    and    are special cases of the above. The form of the result is a sum of trigonometric terms whose arguments are integral linear combinations of the original arguments.

An important special case is when the input is a polynomial in  and  over a field, in which case the result is a canonical form; namely,

where ,  are in the field and  is bounded by the total degree of the input polynomial in  and .