Chapter 7 - Section 7.2 - Probability Theory - Exercises - Page 467: 16

Shown below.

Suppose E and F are independent. Then \begin{align*} P(\bar{E} \cap \bar{F}) &= 1 - P(E \cup F) \\ &= 1 - [P(E) + P(F) - P(E \cap F)] \\ &= 1 - P(E) - P(F) - P(E)P(F) \\ &= (1 - P(E))(1 - P(F)) \\ &= P(\bar{E})P(\bar{F}) \end{align*}