I was playing around with Euler’s infinite product for the sine function, when it occurred to me.
$$\sin(x)=x\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2\pi^2}\right)$$
$$\log(\sin(x))=\log(x)+\sum_{n=1}^{\infty}\log\left(1-\frac{x^2}{n^2\pi^2}\right)$$
Is this a valid series expansion? I assume so. Also, over what interval does it converge?