Laurent series expansion at what point? The function is holomorphic everywhere in $\mathbb{C}\backslash\{2k\pi, k\in\mathbb{Z}\}$. This means that for any $z_0\neq 2 k \pi$, the Laurent series expansion at $z_0$ will have a zero principal part.
As for the Laurent series expansion at any of the singularities, consider for example the expansion at $z_0=0$. Since for $|z|\ll 1$, $\cos z-1 = -\frac{1}{2}z^2 + \frac{1}{24}z^4+O(z^6)$ we have $ \frac{1}{(\cos z-1)^2} = \frac{4}{z^4}\left(1-\frac{1}{12} z^2 + O(z^4)\right)^{-2} = \frac{4}{z^4} + \frac{2}{3 z^2} + O(1) $ So the principal part of the Laurent series expansion at $z=0$ is $\frac{4}{z^4} + \frac{2}{3 z^2}$.