Question

Math

Posted 4 months ago

**Solution:** This geometric series is
\[1 + \cos^2 \theta + \cos^4 \theta + \cdots = \frac{1}{1 - \cos^2 \theta} = 5.\]
Hence,
\[\cos^2 \theta = \frac{4}{5}.\]
Then
\[\cos 2\theta = 2 \cos^2 \theta - 1 = \

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If $\sum_{n=0}^{\infty} \cos ^{2 n} \theta=5$, what is $\cos 2 \theta$ ?
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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

We start with the given series: $\sum_{n=0}^{\infty} \cos^{2n} \theta = 5$

step 2

Recognize that this is a geometric series with the first term $a = 1$ and common ratio $r = \cos^2 \theta$

step 3

The sum of an infinite geometric series is given by $\frac{a}{1 - r}$, so we have $\frac{1}{1 - \cos^2 \theta} = 5$

step 4

Simplify the equation: $1 - \cos^2 \theta = \frac{1}{5}$

step 5

Recognize that $1 - \cos^2 \theta = \sin^2 \theta$, so $\sin^2 \theta = \frac{1}{5}$

step 6

Therefore, $\cos^2 \theta = 1 - \frac{1}{5} = \frac{4}{5}$

step 7

We need to find $\cos 2 \theta$. Recall the double-angle formula: $\cos 2 \theta = 2 \cos^2 \theta - 1$

step 8

Substitute $\cos^2 \theta = \frac{4}{5}$ into the formula: $\cos 2 \theta = 2 \left(\frac{4}{5}\right) - 1 = \frac{8}{5} - 1 = \frac{3}{5}$

Answer

$\cos 2 \theta = \frac{3}{5}$

Key Concept

Geometric Series Sum Formula

Explanation

The sum of an infinite geometric series with first term $a$ and common ratio $r$ (where $|r| < 1$) is given by $\frac{a}{1 - r}$. This formula was used to solve for $\cos^2 \theta$ and subsequently $\cos 2 \theta$.

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