Quadruple Angle Formulas/Sine/Corollary 2
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Corollary to Quadruple Angle Formula for Sine
For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
- $\dfrac {\sin 4 \theta} {\sin \theta} = 2 \cos 3 \theta + 2 \cos \theta$
where $\sin$ denotes sine and $\cos$ denotes cosine.
Proof
First note that when $\theta = 0, \pm \pi, \pm 2 \pi \ldots$:
- $\sin \theta = 0$
so $\dfrac {\sin 4 \theta} {\sin \theta}$ is undefined.
Therefore for the rest of the proof it is assumed that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
| \(\ds \frac {\sin 4 \theta} {\sin \theta}\) | \(=\) | \(\ds 8 \cos^3 \theta - 4 \cos \theta\) | Quadruple Angle Formula for Sine: Corollary 1 | |||||||||||
| \(\ds \) | \(=\) | \(\ds 8 \paren {\dfrac {3 \cos \theta + \cos 3 \theta} 4} - 4 \cos \theta\) | Power Reduction Formulas: Cosine Cubed | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \cos 3 \theta + 2 \cos \theta\) | multiplying out and gathering terms |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $93 \ \text{(a)}$
- (although see Quadruple Angle Formulas/Sine/Mistake for analysis of an error in that work)