Normal form of trigonometric "polynomials" In the context of trigonometry, we often meet terms like 
$\sin(\alpha)\cos(\beta/2) + \tan(\beta)\cos(\alpha)$
which are "polynomials" in $\sin$, $\cos$, $\tan$, and the parameters are rational multiples of angles (like $\alpha$ or $\beta/2$). 
I wonder whether there is well-defined notion of "normal form" of these expressions, in the sense we can compare any two of these expressions by normalising them in a finite series of steps. 
 A: As $E_1=E_2$ if and only if $E_1-E_2=0$, would you settle for the slightly weaker question, "can we reduce such an expression in a routine way to one which we can immediately see whether it is $0$ or not"?
To answer this weaker question, then:
We can at once get rid of all trigonometric functions except $\cos$ and $\sin$. Using the addition formulas we can separate out all the indeterminates, so that we only have to worry about things like $\cos\frac{r}{s}\alpha$ and $\sin\frac{r}{s}\alpha$ where $r,s$ are various natural numbers. 
Now look at all the appearances of the first indeterminate $\alpha$. 
There will be a finite number of denominators, $q_1,\dots,q_k$ of the rational multipliers of $\alpha$. So putting $q=\text{lcm}(q_1,\dots,q_k)$ we only see $\alpha$ in things like $\cos\frac{r}{q}\alpha$. We can use the addition formula to re-express this in terms of $\cos\frac{\alpha}{q}$ and $\sin\frac{\alpha}{q}$. 
Now use the "tangent of half the angles" formulas
$$\cos\theta=\frac{1-t^2}{1+t^2}, \sin\theta=\frac{2t}{1+t^2}\ \text{where}\ t:=\tan\frac{\theta}{2}
$$
to replace all the $\cos\frac{\alpha}{q}$ and $\sin\frac{\alpha}{q}$ by quotients of polynomials in $t_{\alpha}:=\tan\frac{\alpha}{2q}$.
Deal with each of the other indeterminates $\beta,\dots$ in a similar way. 
The whole expression can then be expressed as the quotient of two polynomials $f(t_\alpha,\dots,t_\omega)$ and $g(t_\alpha,\dots,t_\omega)$. 
Our original expression is $0$ if and only if every coefficient of the polynomial $f$ is $0$.
[Not recommended as a way to prove trigonometric identities, unless all else has failed!]
