Construct an equation of the parabola $y^2=2px$ in polar coordinates, where it's vertex is at $(0,0)$.

Question

I got

$\begin{split} &r^{2}\sin^{2}\varphi=2pr\cos\varphi \\& r(r\sin^{2}\varphi-2p\cos\varphi)=0 \\& r\sin^{2}\varphi-2p\cos\varphi=0 \implies r=\frac{2p\cos\varphi}{\sin^{2}\varphi} \end{split}$

So I got it's $\begin{cases}r=\frac{2p\cos\varphi}{\sin^{2}\varphi}\text{ if } \varphi \ne \pi/2 \\ r=0,\text{ if } \varphi = \pi/2 \end{cases}$

I'm not sure if this is correct or not. What bugs me is that r^2, so I can't express r normally and don't know if doing it by cases is correct.