Sine of Integer Multiple of Argument/Formulation 1/Lemma
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Theorem
- For $n \in \Z$:
| \(\ds \map \cos {n \theta} \map \sin {\theta}\) | \(=\) | \(\ds \map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1 } \theta}\) |
Proof
| \(\ds \map \cos {n \theta} \map \sin {\theta}\) | \(=\) | \(\ds \map \cos {n \theta} \map \sin {\theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\map \sin {n \theta} \map \cos {\theta} - \map \sin {n \theta} \map \cos {\theta} } + \map \cos {n \theta} \map \sin {\theta}\) | add zero | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \sin {n \theta} \map \cos {\theta} - \paren {\map \sin {n \theta} \map \cos {\theta} - \map \cos {n \theta} \map \sin {\theta} }\) | regroup | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \sin {n \theta} \map \cos {\theta} - \map \sin {n \theta - \theta}\) | Sine of Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1} \theta}\) | simplification |
$\blacksquare$