2004 AMC 12A Problem 21

If $\sum\limits_{n = 0}^{\infty}\cos^{2n}\theta = 5$, what is the value of $\cos{2\theta}$?

$\text {(A) } \frac15 \qquad \text {(B) } \frac25 \qquad \text {(C) } \frac {\sqrt5}{5}\qquad \text {(D) } \frac35 \qquad \text {(E) }\frac45$

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.