Show that $f(z)$ is meromorphic on $\mathbb{C}\setminus(-\infty,-1]$ and prove that $z=0$ is a simple pole

Question

I'm being asked to show that $\frac{\log(1+z)}{\cos(z)-1}$ is a meromorphic function on $\mathbb{C}\setminus(-\infty,-1]$ as well as to find its simple poles.

My idea here is that I can start by saying that my function is of the form $\frac{g(z)}{h(z)}$ where g,h are entire functions on the domain. Both functions are defined and analytical on $\mathbb{C}\setminus(-\infty,-1]$, therefore it follows that $f(z)$ is meromorphic.

Would this be enough to show that the function is meromorphic? Is there a more rigorous mathematical proof that I can do to prove it?

I would also appreciate any help for me to find its simple poles

Answer 1

Yes, it is a meromorphic function for that reason that you mentioned, assuming that here $\log$ is the analytic branch of the logarithm defined on $\mathbb C\setminus(-\infty,0]$.

And it has a single simple pole, at $0$, because $=$ is a simple root of the numerator and a double root of the donominator. The remaining poles are located at $2k\pi$ ($k\in\mathbb N$) and they are all double poles.