Cosecant expansion near zero ($\csc z=z^{-1}+z/6+7z^3/360+O(z^5)$)

Factor the sine expansion as $\sin z=z[1-z^2/6+z^4/120+O(z^6)]$. To invert the bracket, put $1+A{z}^2+B{z}^4+O(z^6)$ and equate coefficients after multiplication. The $z^2$ coefficient gives $A=1/6$, while the $z^4$ coefficient gives $B=A/6-1/120=7/360$. Dividing by $z$ proves the displayed Laurent series truncation for $z\to 0$, $z\ne 0$. This reciprocal-series calculation is useful when a small sine argument is itself an inverse power of a large variable; multiply the remainder by any outside prefactor when assigning its final order.