Sine to Power of Odd Integer

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Theorem

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


Proof

\(\displaystyle \sin^{2 n + 1} \theta\) \(=\) \(\displaystyle \paren {\frac {e^{i \theta} - e^{-i \theta} } {2 i} }^{2 n + 1}\) Sine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1} } {\paren {2 i}^{2 n + 1} }\) Power of Product
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1} } {\paren {2 i} \paren {2}^{2 n} \paren {i}^{2 n} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1}\) $i^2 = -1$, Exponent Combination Laws/Power of Power
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} 2i } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k e^{\paren {2 n + 1 - k } i \theta} \paren {-1}^{k} e^{-\paren {k} i \theta}\) Binomial Theorem
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} 2i } \sum^{2 n + 1}_{k \mathop = 0} \paren {-1}^{k} \binom {2 n + 1} k e^{\paren {2 n + 1 - 2 k } i \theta}\) Exponential of Sum
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {\sum^n_{k \mathop = 0} \paren {-1}^{k} \binom {2 n + 1} k e^{\paren {2 n + 1 - 2 k } i \theta} + \sum^{2 n + 1}_{k \mathop = n + 1} \paren {-1}^{k} \binom {2 n + 1} k e^{\paren {2 n + 1 - 2 k } i \theta} }\) partitioning the sum


\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \paren {-1}^{2 n + 1 - k} \binom {2 n + 1} {2 n + 1 - k} e^{\paren {2 n + 1 - 2 \paren {2 n + 1 - k } } i \theta} }\) $k \mapsto 2 n + 1 - k$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \paren {-1}^{2 n} \paren {-1}^{1} \paren {-1}^{- k} \binom {2 n + 1} {2 n + 1 - k} e^{\paren {2 n + 1 - 2 \paren {2 n + 1 - k } } i \theta} }\) Exponent Combination Laws/Product of Powers
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} 2i } \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} - \sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} }\) Symmetry Rule for Binomial Coefficients


\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} } \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k \frac {e^{\paren {2 n - 2 k + 1} i \theta} - e^{-\paren {2 n - 2 k + 1} i \theta} } {2i} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} } \sum_{k \mathop = 0}^n \paren {-1}^k \binom {2 n + 1} k \sin \paren {2 n - 2 k + 1} \theta\) Sine Exponential Formulation

$\blacksquare$


Sources