On a one-parameter class of cosine polynomials

We prove: Let $a\geq 0$ be a real number. For any integer $n\geq 2$ and any real $x\in (0,\pi)$, we have
$$ 1+\cos(x)+\sum_{k=2}^n \frac{\cos(kx)}{k+a} >\frac{1}{(a+2)(a+3)}. $$
The lower bound is sharp. This extends a result of Brown and Koumandos, who proved the inequality for the special case $a=0$.