If you multiply both sides by ##x(x-\pi)(x+\pi)/\pi^2##, and note that ##\sin(\pi) = 0## on the right hand side, you will end up with the product formula for the sine function, due to Euler:
$$\sin(x) = x \prod_{n=1}^{\infty}\left(1 - \frac{ x^2}{\pi^2 n^2}\right)$$
He obtained this formula by rather brashly viewing the sine function as a "polynomial" with infinitely many roots, namely ##0, \pm \pi, \pm 2\pi, \pm 3\pi, \ldots##. Accordingly, he "factored" it as follows:
$$\sin(x) = kx(x \pm \pi)(x \pm 2\pi)(x \pm 3\pi)\ldots$$
for some constant ##k##. The constant must be chosen so that ##\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1##, which forces
$$\sin(x) = x\left(1 \pm \frac{x}{\pi}\right)\left(1 \pm \frac{x}{2\pi}\right)\left(1 \pm \frac{x}{3\pi}\right)\ldots$$
Each plus/minus factor can be simplified as follows using the rule ##(a-b)(a+b) = a^2 - b^2##:
$$1 \pm \frac{x}{n\pi} = 1 - \frac{x^2}{n^2 \pi^2}$$
and the result follows.
Of course the above is completely nonrigorous. Euler had the extraordinary ability to turn invalid manipulations into valid results! The same result can be obtained rigorously by using Fourier series. See for example Courant and John, Introduction to Calculus and Analysis I, page 602.
