sequences and series - How to show that the limit $ lim_{n to infty}sin frac{n theta}{2}sin frac{(n+1) theta}{2}$ doesn't exist?

I've been trying to show that $\sum_n \sin n\theta$ diverges (for most thetas at least), and I've come up with this expression for the partial sum (up to a multiplicative constant), and now I want to show that its limit doesn't exist:

$$ \lim_{n \to \infty}\sin \frac{n \theta}{2}\sin \frac{(n+1) \theta}{2}$$

But I don't know how to proceed with this. Both terms are divergent, but that doesn't mean their product necessarily diverges (though in this case it sure seems so). Is there a straightforward way to show this limit doesn't exist?

EDIT: I want to clarify that while I did originally set out to show the divergence of a series, that's not the aim of this question, which is how to rigorously show a limit doesn't exist. I can show that the limit doesn't equal $0$, but I want to learn how to show that it can't equal any other number as well.

Answer

Note that $\sin\alpha\sin\beta=\frac12(\cos(\alpha-\beta)-\cos(\alpha+\beta))$, hence
$$\sin\frac{n\theta}2\sin\frac{(n+1)\theta}2=\frac12\left(\cos\frac\theta2-\cos\bigl((n+\tfrac12)\theta\bigr)\right) $$
so unless $\theta$ is special ...