Cosine to Power of Even Integer

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Theorem

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


Proof 1

\(\ds \cos^{2 n} \theta\) \(=\) \(\ds \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^{2 n}\) Euler's Cosine Identity
\(\ds \) \(=\) \(\ds \frac 1 {2^{2 n} } \paren {e^{i \theta} + e^{-i \theta} }^{2 n}\) Power of Product
\(\ds \) \(=\) \(\ds \frac 1 {2^{2 n} } \sum^{2 n}_{k \mathop = 0} \binom{2 n} k e^{k i \theta} e^{-\paren {2 n - k} i \theta}\) Binomial Theorem
\(\ds \) \(=\) \(\ds \frac 1 {2^{2 n} } \sum^{2 n}_{k \mathop = 0} \binom{2 n} k e^{-\paren {2 n - 2 k} i \theta}\) Exponential of Sum
\(\ds \) \(=\) \(\ds \frac 1 {2^{2 n} } \paren {\sum^{n \mathop - 1}_{k \mathop = 0} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} + \binom {2 n} n e^{\paren {2 n - 2 n} i \theta} + \sum^{2 n}_{k \mathop = n + 1} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} }\) partitioning the sum
\(\ds \) \(=\) \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n} } \paren {\sum^{n - 1}_{k \mathop = 0} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} + \sum^{2 n}_{k \mathop = n + 1} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} }\) Exponential of Zero
\(\ds \) \(=\) \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n} } \paren {\sum^{n - 1}_{k \mathop = 0} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} + \sum^{n - 1}_{k \mathop = 0} \binom {2 n} {2 n - k} e^{\paren {2 \paren {2 n - k} - 2 n} i \theta} }\) $k \mapsto 2 n - k$
\(\ds \) \(=\) \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n} } \paren {\sum^{n - 1}_{k \mathop = 0} \binom {2 n} k e^{-\paren {2 n - 2 k} i \theta} + \sum^{n - 1}_{k \mathop = 0} \binom {2 n} k e^{\paren {2 n - 2 k} i \theta} }\) Symmetry Rule for Binomial Coefficients
\(\ds \) \(=\) \(\ds \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n - 1} } \sum^{n - 1}_{k \mathop = 0} \binom {2 n} k \map \cos {2 n - 2 k} \theta\) Euler's Cosine Identity

$\blacksquare$


Proof 2

\(\ds \cos^n \theta\) \(=\) \(\ds \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^n\) 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 term of the sequence $0, 1, \ldots, n$ is $\dfrac n 2$.

So when $k = \dfrac n 2$ we have $n - 2k = 0$ and $n - k = n - \dfrac n 2 = \dfrac n 2$.

Thus:

\(\ds \binom n k\) \(=\) \(\ds \frac {n!} {\paren {\paren {n / 2}!} \paren {\paren {n - n / 2}!} }\)
\(\ds \) \(=\) \(\ds \frac {n!} {\paren {\paren {n / 2}!}^2}\)


So the middle term collapses to:

\(\ds R_n\) \(=\) \(\ds 2 \frac {n!} {\paren {\paren {n / 2}!}^2} \frac {e^{0 i \theta} } 2\)
\(\ds \) \(=\) \(\ds \frac {n!} {\paren {\paren {n / 2}!}^2}\)

$\blacksquare$


Also defined as

This result is also reported in a less elegant form as:

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

for all even $n$.


Sources