Cosine to Power of Odd Integer/Proof 1

Theorem

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

Proof

\(\ds \cos^{2 n + 1} \theta\)

\(=\)

\(\ds \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^{2 n + 1}\)

Euler's Cosine Identity

\(\ds \)

\(=\)

\(\ds \frac 1 {2^{2 n + 1} } \paren {e^{i \theta} + e^{-i \theta} }^{2 n + 1}\)

Power of Product

\(\ds \)

\(=\)

\(\ds \frac 1 {2^{2 n + 1} } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k e^{k i \theta} e^{-\paren {2 n - k + 1} i \theta}\)

Binomial Theorem

\(\ds \)

\(=\)

\(\ds \frac 1 {2^{2 n + 1} } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta}\)

Exponential of Sum

\(\ds \)

\(=\)

\(\ds \frac 1 {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} + \sum^{2 n + 1}_{k \mathop = n + 1} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} }\)

partitioning the sum

\(\ds \)

\(=\)

\(\ds \frac 1 {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} {2 n + 1 - k} e^{\paren {2 \paren {2 n - k + 1} - 2 n - 1} i \theta} }\)

$k \mapsto 2 n + 1 - k$

\(\ds \)

\(=\)

\(\ds \frac 1 {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} }\)

Symmetry Rule for Binomial Coefficients

\(\ds \)

\(=\)

\(\ds \frac 1 {2^{2 n} } \sum^n_{k \mathop = 0} \binom {2 n + 1} k \map \cos {2 n - 2 k + 1} \theta\)

Euler's Cosine Identity

$\blacksquare$