Answer
$\frac{1}{z} = \frac{1}{r}~(cos~\theta-i~sin~\theta)$
Work Step by Step
$z = r(cos~\theta+i~sin~\theta)$
We can find an expression for $\frac{1}{z}$:
$\frac{1}{z} = \frac{1}{r(cos~\theta+i~sin~\theta)}$
$\frac{1}{z} = \frac{1}{r(cos~\theta+i~sin~\theta)}~\times ~\frac{cos~\theta-i~sin~\theta}{cos~\theta-i~sin~\theta}$
$\frac{1}{z} = \frac{cos~\theta-i~sin~\theta}{r(cos^2~\theta+sin^2~\theta)}$
$\frac{1}{z} = \frac{cos~\theta-i~sin~\theta}{r(1)}$
$\frac{1}{z} = \frac{1}{r}~(cos~\theta-i~sin~\theta)$