Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2004 / Analysis / 5 / c / Solution

An essential singularity is an isolated singularity which is neither removable nor a pole; equivalently its Laurent expansion has infinitely many nonzero negative-power terms.

For the final unnumbered classification, the only possible finite singularities are the zeros of the denominator, namely $z=\pm i$ and $z=k\pi$ for integers $k$. At zero,so zero is removable, with extension value one. At each nonzero $k\pi$, $\sin z$ has a simple zero, the other denominator factor is nonzero and the numerator is nonzero. These points are poles of order two, with leading coefficient $(k\pi {)}^2/((k\pi {)}^2+1)$.

At $z=\pm i$, the polynomial denominator has a simple zero. Since $\sin (\pm i)=\pm i\sinh 1\ne 0$ and the numerator is $-1$, these are simple poles. There are no essential singularities on $\mathbb{C}$ and no other finite singularities.