Cosine to Power of Odd Integer/Proof 2

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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

\(\ds \cos^n \theta\) \(=\) \(\ds \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^n\) De Moivre's Theorem
\(\ds \) \(=\) \(\ds \frac {\paren {e^{i \theta} + e^{-i \theta} }^n} {2^n}\)
\(\ds \) \(=\) \(\ds \frac 1 {2^n} \sum_{k \mathop = 0}^n \binom n k e^{\paren {n - k} i \theta} e^{-k i \theta}\) Binomial Theorem
\(\ds \) \(=\) \(\ds \frac 1 {2^n} \sum_{k \mathop = 0}^n \binom n k e^{\paren {n - 2 k} i \theta}\)


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

\(\ds 2^n \cos^n \theta\) \(=\) \(\ds e^{n i \theta} + \binom n 1 e^{\paren {n - 2} i \theta} + \binom n 2 e^{\paren {n - 4} i \theta} + \cdots\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \binom n {n - 2} e^{-\paren {n - 4} i \theta} + \binom n {n - 1} e^{-\paren {n - 2} i \theta} + e^{-n i \theta}\)
\(\ds \leadsto \ \ \) \(\ds \frac {2 n} 2 \cos^n \theta\) \(=\) \(\ds \paren {\frac {e^{n i \theta} + e^{-n i \theta} } 2} + \binom n 1 \paren {\frac {e^{\paren {n - 2} i \theta} + e^{-\paren {n - 2} i \theta} } 2}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \binom n 2 \paren {\frac {e^{\paren {n - 4} i \theta} + e^{-\paren {n - 4} i \theta} } 2} + \cdots\)


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

\(\ds n - 2 k\) \(=\) \(\ds n - 2 \frac {n - 1} 2\)
\(\ds \) \(=\) \(\ds n - \frac {2 n - 2} 2\)
\(\ds \) \(=\) \(\ds n - n + 1\)
\(\ds \) \(=\) \(\ds 1\)

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

\(\ds n + 2 k\) \(=\) \(\ds n - 2 \frac {n + 1} 2\)
\(\ds \) \(=\) \(\ds n - \frac {2 n + 2} 2\)
\(\ds \) \(=\) \(\ds n - n - 1\)
\(\ds \) \(=\) \(\ds -1\)


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

\(\ds n - \frac {n - 1} 2\) \(=\) \(\ds \frac {2 n - n - 1} 2\)
\(\ds \) \(=\) \(\ds \frac {n + 1} 2\)


So:

\(\ds \binom n {\paren {n - 1} / 2}\) \(=\) \(\ds \frac {n!} {\paren {\frac {n - 1} 2}! \paren {\frac {n + 1} 2}!}\)
\(\ds \) \(=\) \(\ds \binom n {\paren {n + 1} / 2}\)


Thus the two middle terms collapse to:

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

$\blacksquare$


Also defined as

This result is also reported in the form:

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

for all $n \in \Z$.


Sources

(although see Cosine to Power of Odd Integer/Mistake for analysis of an error in that work)