I want to show that there is some $\gamma(n)=o(n^{-1})$ and some $C(n) \to -\infty$ such that for $\gamma \leq \theta \leq \pi$ we have

$\sum_{k=1}^n -1+\cos(k\theta) \leq C(n)$.

If we rewrite this using the Dirichlet kernel, what I want is that: if $\gamma \leq \theta \leq \pi$ then

$-n+\frac{D_n(\theta)}{2}-\frac{1}{2} \leq C(n)$.

If I didn't demand that $\gamma=o(n^{-1})$ then I could do this with $\gamma=n^{-3/4}$ say, and $C(n)$ could then just be $-n$.

I've plotted this a bunch and it looks true, and I've spent the morning trying to prove it. I suspect I may have to not just bound things using absolute values, but actually know where things are positive and negative