Sum of Infinite Series of Product of nth Power of cos 2 theta by 2n+1th Multiple of Sine

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Theorem

Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$.


Then:

\(\ds \sum_{n \mathop = 0}^\infty \paren {\cos 2 \theta}^n \sin \paren {2 n + 1} \theta\) \(=\) \(\ds \sin \theta + \cos 2 \theta \sin 3 \theta + \paren {\cos 2 \theta}^2 \sin 5 \theta + \paren {\cos 2 \theta}^3 \sin 7 \theta + \cdots\)
\(\ds \) \(=\) \(\ds \dfrac {\csc \theta} 2\)


Proof

Let $\theta \ne \dfrac {m \pi} 2$ for any $m \in \Z$.

Then $\size {\cos 2 \theta} < 1$.

\(\ds \sum_{k \mathop \ge 0} \sin \paren {2 k + 1} \theta r^k\) \(=\) \(\ds \dfrac {\paren {1 + r} \sin \theta} {1 - 2 r \cos 2 \theta + r^2}\) Power Series of Sine of Odd Theta: $\size r < 1$
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^\infty \paren {\cos 2 \theta}^k \sin \paren {2 k + 1} \theta\) \(=\) \(\ds \dfrac {\paren {1 + \cos 2 \theta} \sin \theta} {1 - 2 \cos 2 \theta \cos 2 \theta + \paren {\cos 2 \theta}^2}\) setting $r = \cos 2 \theta$
\(\ds \) \(=\) \(\ds \dfrac {\paren {1 + \cos 2 \theta} \sin \theta} {1 - \paren {\cos 2 \theta}^2}\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {\paren {1 + 2 \cos^2 \theta - 1} \sin \theta} {1 - \paren {2 \cos^2 \theta - 1}^2}\) Double Angle Formula for Cosine
\(\ds \) \(=\) \(\ds \dfrac {2 \cos^2 \theta \sin \theta} {1 - \paren {4 \cos^4 \theta - 4 \cos^2 \theta + 1} }\) simplifying and expanding
\(\ds \) \(=\) \(\ds \dfrac {2 \cos^2 \theta \sin \theta} {4 \cos^2 \theta \paren {1 - \cos^2 \theta} }\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {2 \cos^2 \theta \sin \theta} {4 \cos^2 \theta \sin^2 \theta}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \dfrac 1 {2 \sin \theta}\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {\csc \theta} 2\) Definition of Real Cosecant Function

$\blacksquare$


Sources

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