Finding the order of the pole of the complex function $f(z)=\frac{1}{\cos(z)-\sin(z)}$

Question

I am new to complex analysis , this was a example problem and the author just says that as $z=\pi/4$ an isolated singularity , it is clear that the order of the pole is one.

But I am not able to see why ? From what I understood , a pole of order one , means in the Laurent's expansion , the negative term's order is maximum of one . How can one deduce the expansion from just looking at an isolated singularity

Answer 1

The comment of Somos gives an elegant solution, but alternatively, let be $$g(z) = \cos(z) - \sin(z).$$ $\pi/4$ is a zero of order 1 of $g$ because (check yourself) $g(\pi/4) = 0$ and $g'(\pi/4)\ne 0$.