 Solution

From the addition formula and $\sin x_0=0$, $\cos x_0=-1$, so $2x_0$ is a period. Repeatedly subtract $2x_0$ from a positive period $2x_1$; the remainder in $[0,2x_0)$ is also a period, and positivity plus the addition identities force it to be $0$. Thus $x_1=n{x}_0$; the converse follows from periodicity and oddness.