Power Reduction Formulas/Sine Cubed/Proof 1
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Theorem
- $\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$
Proof
| \(\ds \sin 3 x\) | \(=\) | \(\ds 3 \sin x - 4 \sin^3 x\) | Triple Angle Formula for Sine | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 4 \sin^3 x\) | \(=\) | \(\ds 3 \sin x - \sin 3 x\) | rearranging | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sin^3 x\) | \(=\) | \(\ds \frac {3 \sin x - \sin 3 x} 4\) | dividing both sides by $4$ |
$\blacksquare$