Answer
$\displaystyle \frac{ac+bd}{c^{2}+d^{2}}+\frac{ad-bc}{c^{2}+d^{2}}i$
Work Step by Step
$\displaystyle \frac{a-bi}{c-di}\qquad$ ...rationalize by multiplying both the numerator and denominator with $c+di$.
$=\displaystyle \frac{(a-bi)(c+di)}{(c-di)(c+di)}\qquad$ ...use the FOIL method
$=\displaystyle \frac{ac+adi-bci-bdi^{2}}{c^{2}+cdi-cdi-d^{2}i^{2}}\qquad$ ...simplify and add like terms ($i^{2}=-1$).
$=\displaystyle \frac{ac+adi-bci+bd}{c^{2}+d^{2}}\qquad$ ...group the real and imaginary parts in the numerator.
$=\displaystyle \frac{ac+bd+(ad-bc)i}{c^{2}+d^{2}}\qquad$ ...write in standard form $a+bi$
$=\displaystyle \frac{ac+bd}{c^{2}+d^{2}}+\frac{ad-bc}{c^{2}+d^{2}}i$
