We know $\sin(x)=0$ has solutions $0,\pm\pi,\pm2\pi,\pm3\pi,\dots$.
So $\sin(x)$, if interpreted as a polynomial, could be written as:
$a_0x^0+a_1x^1+a_2x^2+\cdots$ and we know this polynomial too:
$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$
So, the question is, is it possible to transform the factored form of $\sin(x)$:
$\sin(x)=a x(x-\pi)(x+\pi)(x-2\pi)(x+2\pi)(x-3\pi)(x+3\pi)\dots$
to
$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots\ ?$