Cosine to Power of Even Integer/Proof 1
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Theorem
| \(\ds \cos^{2 n} \theta\) | \(=\) | \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n - 1} } \paren {\cos 2 n \theta + \binom {2 n} 1 \map \cos {2 n - 2} \theta + \cdots + \binom {2 n} {n - 1} \cos 2 \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n - 1} } \sum_{k \mathop = 0}^{n - 1} \binom {2 n} k \map \cos {2 n - 2 k} \theta\) |
Proof
| \(\ds \cos^{2 n} \theta\) | \(=\) | \(\ds \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^{2 n}\) | Euler's Cosine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \paren {e^{i \theta} + e^{-i \theta} }^{2 n}\) | Power of Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \sum^{2 n}_{k \mathop = 0} \binom{2 n} k e^{k i \theta} e^{-\paren {2 n - k} i \theta}\) | Binomial Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \sum^{2 n}_{k \mathop = 0} \binom{2 n} k e^{-\paren {2 n - 2 k} i \theta}\) | Exponential of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \paren {\sum^{n \mathop - 1}_{k \mathop = 0} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} + \binom {2 n} n e^{\paren {2 n - 2 n} i \theta} + \sum^{2 n}_{k \mathop = n + 1} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} }\) | partitioning the sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n} } \paren {\sum^{n - 1}_{k \mathop = 0} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} + \sum^{2 n}_{k \mathop = n + 1} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} }\) | Exponential of Zero | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n} } \paren {\sum^{n - 1}_{k \mathop = 0} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} + \sum^{n - 1}_{k \mathop = 0} \binom {2 n} {2 n - k} e^{\paren {2 \paren {2 n - k} - 2 n} i \theta} }\) | $k \mapsto 2 n - k$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n} } \paren {\sum^{n - 1}_{k \mathop = 0} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} + \sum^{n - 1}_{k \mathop = 0} \binom {2 n} k e^{\paren {2 n - 2 k} i \theta} }\) | Symmetry Rule for Binomial Coefficients | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n - 1} } \sum^{n - 1}_{k \mathop = 0} \binom {2 n} k \map \cos {2 n - 2 k} \theta\) | Euler's Cosine Identity |
$\blacksquare$