How can we find the first few terms of the Laurent series of $f(w)={1\over \cos w-1}$ where $w\in \mathbb C, |w|<2\pi$? I am wondering if there is another way maybe exploiting one of the more familiar series expansions instead of brute force formula $c_n={1\over 2\pi i}\int_\gamma {f(w)\over (w-a)^{n+1}}dw$?
Also would it be different if we choose to have $|w|\in (2\pi,4\pi)$ instead?