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The simplify(expr,trig) calling sequence simplifies trigonometric expressions by applying the trigonometric identities ${\sin \left(x\right)}^2+{\cos \left(x\right)}^2=1$ and ${\cosh \left(x\right)}^2-{\sinh \left(x\right)}^2=1$.

If the input is a polynomial in $\sin \left(x\right)$ and $\cos \left(x\right)$ then simplify/trig factors out powers of $\sin \left(x\right)$ and $\cos \left(x\right)$ and applies the identity ${\sin \left(x\right)}^2+{\cos \left(x\right)}^2=1$ to what is left so that the degree of what is left in $\sin \left(x\right)$ is at most 1. Thus the result is of the form:

where $A$ and $B$ are polynomials in $\cos \left(x\right)$. If the input is a polynomial in $\sinh \left(x\right)$ and $\cosh \left(x\right)$ then simplify/trig yields a similar result using the identity ${\cosh \left(x\right)}^2-{\sinh \left(x\right)}^2=1$.

To apply the identity to reduce the polynomial so that the degree in $\sin \left(x\right)$ is at most 1, use the command

If the input involves multiple angles that are integer multiples of each other, for example, $\sin \left(x\right)$, $\sin \left(2x\right)$, and $\cos \left(\frac{x}{2}\right)$ then the trigonometric functions are expressed in terms of a common angle, in this case $\frac{x}{2}$.

If the input is a rational expression in $\sin \left(x\right)$ and $\cos \left(x\right)$ then an algorithm is used to put it in the form $\frac{N}{\mathrm{D}}$ and reduce $N$ and $\mathrm{D}$ to lowest terms such that the total degree of the numerator $N$ (in $\sin \left(x\right)$ and $\cos \left(x\right)$) plus the total degree of the denominator is minimized. In particular, any common factor between $N$ and $\mathrm{D}$ has been cancelled out.

Note: Maple does not rationalize the denominator, that is, write the expression in the form $\frac{A}{B}\sin \left(x\right)+\frac{C}{\mathrm{D}}$ for polynomials $A$, $B$, $C$, and $\mathrm{D}$ in $\cos \left(x\right)$ because this form usually leads to a result that is larger in total degree.

$\mathrm{simplify}\left({\sin \left(x\right)}^2+{\cos \left(x\right)}^2\,\mathrm{trig}\right)$

$1$

$\mathrm{simplify}\left({\cosh \left(x\right)}^2-{\sinh \left(x\right)}^2\,\mathrm{trig}\right)$

$1$

$\mathrm{simplify}\left(\cos \left(2x\right)+{\sin \left(x\right)}^2\,\mathrm{trig}\right)$

${\cos \left(x\right)}^2$

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

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

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

$\cos \left(x\right)\sin \left(x\right)\left(\sin \left(x\right)+\cos \left(x\right)\right)$

$r≔\frac{1-{\cos \left(x\right)}^2+\sin \left(x\right)\cos \left(x\right)}{\sin \left(x\right)\cos \left(x\right)+{\cos \left(x\right)}^2}$

$r≔\frac{\sin \left(x\right)\cos \left(x\right)-{\cos \left(x\right)}^2+1}{\sin \left(x\right)\cos \left(x\right)+{\cos \left(x\right)}^2}$

$\mathrm{simplify}\left(r\,\mathrm{trig}\right)$

$\tan \left(x\right)$

$\mathrm{simplify}\left(\left[{\sin \left(x\right)}^3\,{\sin \left(x\right)}^4\right]\,\mathrm{trig}\right)$

$\left[{\sin \left(x\right)}^3\,{\sin \left(x\right)}^4\right]$

$\mathrm{simplify}\left(\left[{\sin \left(x\right)}^3\,{\sin \left(x\right)}^4\right]\,\left\{{\sin \left(x\right)}^2+{\cos \left(x\right)}^2-1\right\}\,\left[\sin \left(x\right)\right]\right)$

$\left[-\sin \left(x\right){\cos \left(x\right)}^2+\sin \left(x\right)\,{\cos \left(x\right)}^4-2{\cos \left(x\right)}^2+1\right]$

$\mathrm{expand}\left(\sin \left(4x\right)\right)$

$8\sin \left(x\right){\cos \left(x\right)}^3-4\sin \left(x\right)\cos \left(x\right)$

$\mathrm{combine}\left({\sin \left(x\right)}^4\,\mathrm{trig}\right)$

$\frac{3}{8}+\frac{\cos \left(4x\right)}{8}-\frac{\cos \left(2x\right)}{2}$