$\csc z$ has principal part $1/z$ at $0$, and $\csc z-1/z=z/6+7z^3/360+O(z^5)$ is odd and holomorphic on $\abs{z}<\pi$.Suggest a slogan
For \(0<\abs{z}<\pi\), \[ {1\over \sin(z)} = \frac{1}{z}+\frac{1}{3 !} z+\frac{7}{360} z^{3}+O\qty{z^{5}}. \]
What is the Laurent expansion of $1/\sin(z)$ at the origin?