Sine to Power of Odd Integer

Theorem

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

Proof

 $\displaystyle \sin^{2 n + 1} \theta$ $=$ $\displaystyle \paren {\frac {e^{i \theta} - e^{-i \theta} } {2 i} }^{2 n + 1}$ $\quad$ Sine Exponential Formulation $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {\paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1} } {\paren {2 i}^{2 n + 1} }$ $\quad$ Exponent Combination Laws: Power of Product $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {e^{-i \theta} - e^{i \theta} }^{2 n + 1}$ $\quad$ Exponent Combination Laws: Power of Power $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k e^{-k i \theta} \paren {-1}^{2 n - k + 1} e^{\paren {2 n - k + 1} i \theta}$ $\quad$ Binomial Theorem, Exponent Combination Laws: Power of Product $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta}$ $\quad$ Exponent Combination Laws: Product of Powers $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta} + \sum^{2 n + 1}_{k \mathop = n + 1} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta} }$ $\quad$ Partitioning the sum $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} {2 n + 1 - k} \paren {-1}^k e^{\paren {2 n - 2 paren {2 n + 1 - k} + 1} i \theta} }$ $\quad$ $k \mapsto 2n+1-k$ $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^k e^{-\paren {2 n - 2 k + 1} i \theta} }$ $\quad$ Symmetry Rule for Binomial Coefficients $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {-\sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^k e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^k e^{-\paren {2 n - 2 k + 1} i \theta} }$ $\quad$ Exponent Combination Laws: Product of Powers, $\paren {-1}^{2 n} = 0$ $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {\paren {-1}^n } {2^{2 n} } \sum^n_{k \mathop = 0} \binom {2 n + 1} k \map \sin {2 n - 2 k + 1} \theta$ $\quad$ Sine Exponential Formulation $\quad$

$\blacksquare$