TITLE: Find all such polynomials
QUESTION [5 upvotes]: Find all polynomials $P(x)$ with real coefficients, such that,    
$$ |P(x)| \leq \sum_{r=0}^{\infty} |\sin (x)|^{r} \quad \forall x\in\mathbb{R}$$    
This question looks daunting. Please help.

REPLY [2 votes]: In addition to the good and correct answer by Hagen von Eitzen, let me point out that the series is a geometric one and can be evaluated easily:
$$
\sum_{r=0}^{\infty} |\sin (x)|^{r}
=
\begin{cases}
\frac1{1-|\sin(x)|},&\text{when }|\sin(x)|<1\\
\infty,&\text{when }|\sin(x)|=1.
\end{cases}
$$
The sum of the series is $1$ when $x$ is an integer multiple of $\pi$, and that simple case is enough to draw the conclusion, but even more information is available if you want to use it.