(i) Let $x \in \mathbb{R} \setminus \{2j\pi: j \in \mathbb{Z}\}$. Note that $ \sum_{k=1}^n \cos kx=-{1 \over 2}+{{\sin\left(n+{1 \over 2}\right)x} \over {2\sin{x \over 2}}} $ which implies that $ \left|\sum_{k=1}^n \cos kx\right| \leq {1 \over 2}+{1 \over {2\left|\sin{x \over 2}\right|}}. $ In particular, the sequence $\left(\sum_{k=1}^n \cos kx\right)_{n \in \mathbb{N}}$ is bounded. On the other hand, the series $ \sum_{n=2}^\infty \left|{{(-1)^n} \over {n^2-1}}-{{(-1)^{n+1}} \over {(n+1)^2-1}}\right| $ converges by the comparison test, since $ \left|{{(-1)^n} \over {n^2-1}}-{{(-1)^{n+1}} \over {(n+1)^2-1}}\right|=\left|{{2n^2+2n-1} \over {n^4+2n^3-n^2-2n}}\right| \leq {{5n^2} \over {n^4}}={5 \over {n^2}} \qquad (n \gg 1). $ Hence, Dirichlet's test for convergence implies that the series $\sum_{n=2}^\infty (-1)^n {{\cos nx} \over {n^2-1}}$ converges.
(ii) Let $x=2j\pi$ for some $j \in \mathbb{Z}$. Since $\cos nx=1$ for all $n \in \mathbb{N}$, the series $ \sum_{n=2}^\infty (-1)^n {{\cos nx} \over {n^2-1}}=\sum_{n=2}^\infty {{(-1)^n} \over {n^2-1}} $ converges by the Leibniz's alternating series test.