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  ${\sin \left(a\right)}^2$  and  ${\cos \left(b\right)}^2$  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 $\sin \left(x\right)$ and $\cos \left(x\right)$ over a field, in which case the result is a canonical form; namely,

where $a_i$, $b_i$ are in the field and $n$ is bounded by the total degree of the input polynomial in $\sin \left(x\right)$ and $\cos \left(x\right)$.

$\mathrm{combine}\left({\sin \left(x\right)}^2\,\mathrm{trig}\right)$

$\frac{1}{2}-\frac{\cos \left(2x\right)}{2}$

$\mathrm{combine}\left(\sinh \left(x\right)\cosh \left(x\right)\,\mathrm{trig}\right)$

$\frac{\sinh \left(2x\right)}{2}$

$\mathrm{combine}\left(2\sin \left(x\right)\cos \left(y\right)\,\mathrm{trig}\right)$

$\sin \left(x+y\right)+\sin \left(x-y\right)$

$f≔{\sin \left(x\right)}^2\cos \left(x\right)+3{\cos \left(x\right)}^3+2\sin \left(x\right)\cos \left(x\right)$

$f≔{\sin \left(x\right)}^2\cos \left(x\right)+3{\cos \left(x\right)}^3+2\sin \left(x\right)\cos \left(x\right)$

$\mathrm{combine}\left(f\,\mathrm{trig}\right)$

$\frac{5\cos \left(x\right)}{2}+\frac{\cos \left(3x\right)}{2}+\sin \left(2x\right)$

$f≔512{\sin \left(x\right)}^5{\cos \left(x\right)}^5$

$f≔512{\sin \left(x\right)}^5{\cos \left(x\right)}^5$

$\mathrm{combine}\left(f\,\mathrm{trig}\right)$

$\sin \left(10x\right)+10\sin \left(2x\right)-5\sin \left(6x\right)$