I need to prove that
$$arg\left(\frac{z}{w}\right)=arg(z)-arg(w)$$
However, I am a little stuck as to how to go about this.

I know the proof for $arg(zw)=arg(z)+arg(w)$ happens by letting $z=r(cos\theta+isin\theta)$ and $w=s(cos\phi+isin\phi)$ and then multiplying them together and expanding out the brackets, combining the arguments using the double angle formulae.

I have been told that the proof of this is analogous, but I have absolutely no idea how I would simplify
$$\frac{r(cos\theta+isin\theta)}{s(cos\phi+isin\phi)}$$

Is there some identity I don't know about?

EDIT: We have not covered $e^{ix}=cos(x)+isin(x)$ yet, so the proof will not need to use that fact.