Quintuple Angle Formulas/Sine/Proof 1

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Theorem

$\sin 5 \theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta$


Proof

\(\ds \sin 5 \theta\) \(=\) \(\ds \map \sin {3 \theta + 2 \theta}\)
\(\ds \) \(=\) \(\ds \sin 3 \theta \cos 2 \theta + \cos 3 \theta \sin 2 \theta\) Sine of Sum
\(\ds \) \(=\) \(\ds \paren {3 \sin \theta - 4 \sin^3 \theta} \cos 2 \theta + \paren {4 \cos^3 \theta - 3 \cos \theta} \sin 2 \theta\) Triple Angle Formulas
\(\ds \) \(=\) \(\ds \paren {3 \sin \theta - 4 \sin^3 \theta} \paren {\cos^2 \theta - \sin^2 \theta} + \paren {4 \cos^3 \theta - 3 \cos \theta} 2 \sin \theta \cos \theta\) Double Angle Formulas
\(\ds \) \(=\) \(\ds \paren {3 \sin \theta - 4 \sin^3 \theta} \paren {\cos^2 \theta - \sin^2 \theta} + 8 \cos^4 \theta \sin \theta - 6 \cos^2 \theta \sin \theta\) multiplying out
\(\ds \) \(=\) \(\ds \paren {3 \sin \theta - 4 \sin^3 \theta} \paren {1 - 2 \sin^2 \theta} + 8 \paren {1 - \sin^2 \theta}^2 \sin \theta - 6 \paren {1 - \sin^2 \theta} \sin \theta\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds 5 \sin \theta - 20 \sin^3 \theta + 16 \sin ^5 \theta\) multiplying out and gathering terms

$\blacksquare$