Showing weak convergence for $sin^2(nx)$ and determining its limit distribution

Question

I am currently struggling a bit with showing weak convergence for $\sin^2(nx)$ and its limit distribution.

My solution so far, with $\phi \in D(\mathbb{R})$;

$$\left<\sin^2(nx),\phi(x)\right> = \int_{\mathbb R} \sin^2(nx) \phi(x) dx$$

$$=\int_{\mathbb R} \frac{1}{2}(1-\cos(2nx)) \phi(x)$$

I'm stuck at this point.

Possible ideas to solve it:

$\bullet$ Partial integration to get out a $\frac{1}{2n}$ and let $n \to \infty$ and see what happens

$\bullet$ Use triangle inequality and approximate $\cos(2nx)$ to get rid of trigonometric terms.

Would someone like to give a hint what's next in this solution?