Cosine to Power of Even Integer

Theorem

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

That is:


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

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

for all even $n$.