Trigonometry

Let  us now consider some trigonometric expressions. For example:

$$\cos^5 x + \sin^4 x + 2 \cos^2 x - 2 \sin^2 x - \cos 2x = \cos^5 x + \cos^4 x$$

When you type this expression into Maxima, Maple or Mathematica, you must remember that $\cos^5 x$ is really a shorthand for $\left(\cos(x)\right)^5$. So let us try to obtain the above result in Maxima first: 

The function trigexpand

works on expressions such as $\cos{2 x}$and $\sin{\left( \alpha + \beta\right)}$. In turn function trigsimp employs the identities $\sin^2{x} + \cos^2{x} = 1$ and $\cosh^2{x} - \sinh^2{x} = 1$ to simplify expressions containing $\tan$, $\sec$, etc. to $\sin$, $\cos$, $\sinh$, $\cosh$. Further simplification, if needed, may be obtained by using function trigreduce on the result.^2.4

In Maple we don't have to muck around with specialised functions for trigonometry. Maple has a general purpose wrapper called simplify, which recognises the nature of the expression and applies appropriate rules until the result is as compact and simple as possible:

Mathematica will do something quite different here with its own version of Simplify:

We can use Maxima in order to check that

$$2 \cos^2{\frac{x}{2}} \cos^4{x} = \cos^5{x} + \cos^4{x}$$

Function trigrat

gives a canonical  simplified quasilinear form of a trigonometrical expression.^2.5

Mathematica also has similar functions, called TrigExpand, TrigFactor, and TrigReduce, but its TrigExpand doesn't work the same way as in Maxima: