AHSME 1983 Problem 20

If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - px + q = 0$, and $\cot{\alpha}$ and $\cot{\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily $\textbf{(A)} \ pq \textbf{(B)} \ \frac{1}{pq} \textbf{(C)} \ \frac{p}{q^2} \textbf{(D)}\ \frac{q}{p^2} \textbf{(E)}\ \frac{p}{q}$