Sine to Power of Odd Integer

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

$\blacksquare$