Problem 380: (2004 AMC12A, 21)
If $\sum_{n = 0}^{\infty}\cos^{2n}\theta = 5$, 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$
Solution
- The series is geometric with ratio $\cos^2\theta$, summing to $\frac1{1-\cos^2\theta}=\frac1{\sin^2\theta}=5$.
- $\sin^2\theta=\frac15$, so $\cos2\theta=1-2\sin^2\theta=1-\frac25=\frac35$.
- The answer is (D) $\frac35$.
If $\sum_{n=0}^\infty\cos^{2n}\theta=8$, what is the value of $\cos2\theta$?
