Multiplicity of $z=0$ on $f(z)=z\cos(z)-\sin(z)$

Question

I'm trying to find the multiplicity of $z=0$ on $f(z)=z\cos(z)-\sin(z)$ using complex analysis.

I'm new to complex analysis and the argument principle/Rouché's theorem so I'm not quite sure where to start. I can prove how many zero's this function has but I'm not quite sure what theorem would help me determine the multiplicity of this root, can someone point me in the right direction or provide a proof?

Thank you

Answer 1

$$z\cos z -\sin z = z(1-z^2/2+z^4/24+...)-z+z^3/6-z^5/120-....)=$$

$$-z^3/3+z^5/{30} +...$$ Thus it is a zero of multiplicity $3$

Answer 2

Hint!

Use the expansion in power series of $\cos z$ and $\sin z$. the order of $z=0$ is the order of the power series, i.e. the degree of the lowest degree term.