Cosine to Power of Odd Integer

Theorem

 * $\displaystyle \cos^{2n+1} \theta = \frac 1 {2^{2n}} \left({\cos \left({2n+1}\right) \theta + \binom{2n+1} 1 \cos \left({2n-1}\right) \theta + \cdots + \binom{2n+1} n \cos \theta}\right)$

That is:
 * $\displaystyle \cos^{2n+1} \theta = \frac 1 {2^{2n}} \sum_{k \mathop = 0}^n \binom {2n+1} k \cos \left({2n - 2k + 1}\right) \theta$

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

for all odd $n$.