Cosine to Power of Odd Integer/Proof 2

Proof
Matching up terms from the beginning of this expansion with those from the end:

Thus:


 * $\cos^n \theta = \dfrac 1 {2^{n - 1} } \paren {\cos n \theta + n \cos \paren {n - 2} \theta + \dfrac {n \paren {n - 1} } {2!} \cos \paren {n - 4} \theta + \cdots + R_n}$

Now to determine $R_n$.

The middle two terms of the sequence $0, 1, \ldots, n$ are $\dfrac {n - 1} 2$ and $\dfrac {n + 1} 2$.

Thus, when $k = \dfrac {n - 1} 2$:

Similarly, when $k = \dfrac {n + 1} 2$:

The binomial coefficient in each case is the same, because:

So:

Thus the two middle terms collapse to:

Also defined as
This result is also reported in the form:
 * $\displaystyle \cos^{2 n + 1} \theta = \frac 1 {2^{2 n} } \sum_{k \mathop = 0}^n \binom {2 n + 1} k \, \map \cos {2 n - 2 k + 1} \theta$

for all $n \in \Z$.