I am trying to find and classify the singularities of the following :
$$\frac{1}{\cos(Z) + \sin(Z)}$$
and I am not sure how to approach it, I understand the denominator must be zero but I am not sure when that occurs and thus how to classify the singularities.
If $\cos z=0$ the $\cos z+\sin z$ cannot be $0$. If $\cos z \neq 0$ then $\cos z+\sin =0$ iff $\tan z = -1$ which gives $z=-\frac {\pi} 4+2n \pi $ where $n$ is an integer or $z=\frac {3\pi} 4+2n \pi $ where $n$ is an integer.
At each of these points $z_0$ we have $\frac {z-z_0} {\cos z+\sin z} \to \frac 1 {2\cos z_0}$ so these points are simple poles.