Is $\textrm{span}\{\sin(nx): n \in \mathbb N\}$ a sub-algebra of $C([0,\pi])$?

Question

I'm working on a proof that $B=\{\sin(nx):n \in \mathbb N\}$ is an orthonormal basis of $L^2([0,\pi])$ and my missing link is that the sub-algebra of $C([0,\pi])$ generated by $B$ is its span. I've shown it is true for the cosine basis since $$\cos(a)\cos(b)=\frac {\cos(a+b)+\cos(a-b)} 2, $$ but the equivalent formula for $\sin(b)\sin(b)$ also involves cosines. Is the statement I want to prove really true? And if so how do I prove it?