Cosecant ($\csc z$)

The cosecant is $\csc z=1/\sin z$, wherever the sine denominator is nonzero. The square of the cosecant occurs in the derivative $d(\cot z)/dz=-{\csc }^{2}z$ of the cotangent.

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.