<s>[INST] Derivatives of trig polynomials do not increase degree? Let $c = \cos x$ and $s = \sin x$,
and consider a trigonometric polynomial $p(x)$
in $c$ and $s$.
The degree of $p(x)$ is the maximum of
$n+m$ in terms $c^n s^m$.

Is it the case that repeated derivatives of $p(x)$,
  expressed again in terms of $c$ and $s$,
  never increase the degree?

For example, $p(x)=c^4 s^2$ has degree $6$, and
here are its first $5$ derivatives.
$$
c^4 s^2 \\
d^1 = 2 c^5 s-4 c^3 s^3 \\
d^2 = 2 c^6-22 c^4 s^2+12 c^2 s^4 \\
d^3 = -56 c^5 s+136 c^3 s^3-24 c s^5 \\
d^4 = -56 c^6+688 c^4 s^2-528 c^2 s^4+24 s^6 \\
d^5 = 1712 c^5 s-4864 c^3 s^3+1200 c s^5
$$
Because $d^3$ and $d^5$ has the same
$c^5 s + c^3 s^3+ c s^5$ structure, we are in a loop,
establishing that all derivatives of $p(x)$ have degree $6$.
 [/INST] Using complex exponential relations, 
$$e^{i\theta} = \cos\theta + i \sin\theta \qquad
\sin\theta = \frac{1}{2i}(e^{i\theta}-e^{-i\theta}) \qquad 
\cos\theta = \frac{1}{2}(e^{i\theta}+e^{-i\theta})$$
we see that sine-cosine polynomial of degree $p$ can be written as a linear combination of complex exponentials "as large as" $e^{ip\theta}$ and "as small as" $e^{-ip\theta}$. Differentiation preserves these exponentials, each of which can then be re-written as $\cos p\theta + i \sin p\theta$ (and smaller multiples of $\theta$); finally, each multiple-angle trig term expands to a degree-$p$-or-smaller polynomial in $\sin\theta$ and $\cos\theta$. $\square$
Note that expressing everything in terms of complex exponentials neatly avoids the problem @Mathmo123 outlined, where a sine-cosine polynomial's apparent degree may not take into account possible reductions.
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