$r(\theta) e^{i\theta}$ - Parametrization of the square $\gamma$

Question

In the question Contour integral - Circle instead of a square achille hui explains that $r(\theta) e^{i\theta}$ with $r(\theta)\min\left(\frac{1}{|\cos\theta|}, \frac{1}{|\sin\theta|}\right)$ is a suitable choice to get a square. Is anyone could explain to me why this parametrization is correct for how $\gamma$ is defined?

Answer 1

Hint the line $x = 1$ in polar coordinates is $r\cos\theta =1$.