Cosine to Power of Odd Integer

Theorem

 * $\ds \cos^{2 n + 1} \theta = \frac 1 {2^{2 n} } \paren {\cos \paren {2 n + 1} \theta + \binom {2 n + 1} 1 \cos \paren {2 n - 1} \theta + \cdots + \binom {2 n + 1} n \cos \theta}$

That is:
 * $\ds \cos^{2 n + 1} \theta = \frac 1 {2^{2 n} } \sum_{k \mathop = 0}^n \binom {2 n + 1} k \cos \paren {2 n - 2 k + 1} \theta$

Also defined as
This result is also reported in a less elegant form as:
 * $\ds \cos^n \theta = \frac 1 {2^{n - 1} } \sum_{k \mathop = 0}^{\paren {n - 1} / 2} \paren {\binom n k \cos \paren {n - 2 k} \theta}$

for all odd $n$.