How to prove that the convergence of $\sum_{n=1}^{\infty} \frac{\sec^a n}{n^c}$ implies that of $\sum_{n=1}^{\infty} \frac{\csc^a n}{n^c}$

Question

The most general thing I've gotten is that the absolute convergence of $$\sum_{n=1}^{\infty} \frac{\csc^a (n + x)}{n^c}$$ implies that of $$\sum_{n=1}^{\infty} \frac{\csc^a \left(\frac{m}{2} n + \frac{x}{2} + \frac{\pi}{4} \pm \frac{\pi}{4}\right)}{n^c}$$ for integral $m$ and real $x$, but even this isn't enough to prove the theorem. It can easily be proven in the other direction, but I can't seem to prove that the absolute convergence of the secant series implies that of the cosecant series. Any ideas?