
Z_{1}=z_{2}
Suppose $\displaystyle z_1=r_1(\cos(\theta_1)+i\sin(\theta_1))$ and $\displaystyle z_2=r_2(\cos(\theta_2)+i\sin(\theta_2))$. If $\displaystyle z_1=z_2$, then how are $\displaystyle r_1 \ \mbox{and} \ r_2$ related? How are $\displaystyle \theta_1 \ \mbox{and} \ \theta_2$ related?
If $\displaystyle z_1=z_2$, then $\displaystyle a_1=a_2 \ \mbox{and} \ b_1=b_2$. Therefore, $\displaystyle r_1=r_2$ and $\displaystyle \theta_1=\theta_2$ since $\displaystyle a^2+b^2=r^2 \ \mbox{and} \ \theta=\tan\frac{b}{a}$.
Is this correct?

I don't think your answer takes into account fully the ambiguities inherent in polar coordinates. You have negative radii possibilities, as well as changing the angle by certain welldefined amounts. How does this change your picture?

But by definition, $\displaystyle z_1=z_2$ iff. $\displaystyle a_1=a_2 \ \mbox{and} \ b_1=b_2$. Why can't I use that fact?

Sure, but that's cartesian coordinates. As soon as you start talking about radii and angles (i.e., polar coordinates), the game is different. Example:
$\displaystyle 2\cos(\pi/4)=2\cos(5\pi/4).$
In fact, in polar coordinates, the point $\displaystyle 2\,\text{cis}\,\pi/4$ is the same point as $\displaystyle 2\,\text{cis}\,5\pi/4.$
So from this, you can see that setting
$\displaystyle r_{1}\cos(\theta_{1})=r_{2}\cos(\theta_{2})$ is simply not going to imply that $\displaystyle r_{1}=r_{2}$ and $\displaystyle \theta_{1}=\theta_{2}.$
Even if you throw in the other equation (the y equation), you're not going to resolve all the ambiguities.