Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the acute angles that the lines $d_{1}$ and $d_{2}$ shown make with the $x$-axis are $A$ and $B$ respectively, as shown in the figure.
Accordingly, which of the following is the expression for the ratio $\frac{a}{b}$ in terms of $A$ and $B$?
A) $\frac{\tan A}{\tan B}$
B) $\cot A \cdot \cot B$
C) $\cot A - \tan B$
D) $1 + \cot A \cdot \tan B$
E) $1 - \tan A \cdot \cot B$