The transcendence of sine function Are there polynomials $$R_2(x) ,R_1(x),R_0(x)$$ $$R_2(x)\text{ not the zero polynomial} $$ such  as  $$R_2(x)\sin^2(x) +R_1(x)\sin(x)+R_0(x)=0\quad \forall x\in[a,b]?$$
 A: Notice that we are dealing with analytic functions here. Therefore, by the identity theorem, if the equality$$R_2(x)\sin^2(x)+R_1(x)\sin(x)+R_0(x)=0$$holds on $[a,b]$, it holds for every $x\in\mathbb R$.
So, the answer is negative, because then$$(\forall n\in\mathbb{Z}):R_0(n\pi)=-R_2(n\pi)\sin^2(n\pi)-R_1(n\pi)\sin(n\pi)=0,$$Since a non-zero polynomial cannot have infinitely many zeros, $R_0(x)\equiv0$. So, we have$$R_2(x)\sin^2(x)-R_1(x)\sin(x)=0,$$and this implies that $R_2(x)\sin(x)+R_1(x)=0$. By the same argument as above, $R_1(x)\equiv0$. So, $R_2(x)\sin(x)=0$ and this implies that $R_2(x)\equiv0$, which goes against our assumptions.
A: Hint:
Let
$$f(x):=p\cos t+q\sin t+r=0$$ where $p,q,r$ are polynomials.
Differentiating twice and adding the initial equation,
$$f''(t)+f(t)=p''\cos t-2p'\sin t+q''\sin t+2q'\cos t+r+r''=0.$$
In this expression, the degrees of the factors of $\cos t$ and $\sin t$ have decreased by one. We can iterate and in the end we will obtain an expression of the form
$$a\cos t+b\sin t+s=0$$ where at least one of $a,b$ is a nonzero constant. Then by further differentiations,
$$c\cos t+d\sin t=0,$$ which is not possible.
I am fairly confident that this generalizes to quadratic expressions in $\cos t$ and $\sin t$.

Solution:
If you apply the differential operator $(D^4+5D^2+4)$ to an expression of the form
$$p\cos(2t)+q\sin(2t)+r\cos(t)+s\sin(t)+u$$ where at least one of $p,q,r,s$ is a non-constant polynomial, you get an expression of the same form, where the largest polynomial degree is lower.
(For instance, $x^4\cos(2x)\to48x^3\sin(2x)-228x^2\cos(2x)-192x\sin(2x)+24\cos(2x)$. It can be proven formally for any monomial, then polynomial, but the computation is a little tedious.)
By iterating, you eventually obtain a non-trivial linear combination of
$$\cos(2t),\sin(2t),\cos(t),\sin(t),$$
which cannot be identically zero.
