Residue of $f(z) = \frac{z}{1-\cos(z)}$ at $z=2k\pi$

Question

I need to find residue of function $f(z) = \frac{z}{1-\cos(z)}$ at $z=2\pi k$, where $k\in \Bbb Z$. I know residue at $z=0$ from here. I got a hint that need to substitute $z=\hat z+2\pi k$, so $\hat z=z-2\pi k$, so I have to find residue at $\hat z=0$. So $f(\hat z)=\frac{\hat z+2\pi k}{1-\cos(\hat z)}$. One more hint - I have to split function in two parts, when k is odd and k is even.

Answer 1

no it has pole of order $2$ for $2k\pi$.see image.

Answer 2

Hint: As the denominator 1-cosz has a simple pole at z=2kπ, Res(f;2kπ)= lim┬(z→2kπ)⁡〖z/(d/dz(1-cosz))〗