Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.3 The Product and Quotient Theorems - 8.3 Exercises - Page 371: 42

$\frac{1}{z} = \frac{1}{r}~(cos~\theta-i~sin~\theta)$

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