Power Reduction Formulas/Cosine Cubed
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Theorem
- $\cos^3 x = \dfrac {3 \cos x + \cos 3 x} 4$
where $\cos$ denotes cosine.
Proof
\(\ds \cos 3 x\) | \(=\) | \(\ds 4 \cos^3 x - 3 \cos x\) | Triple Angle Formula for Cosine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4 \cos^3 x\) | \(=\) | \(\ds 3 \cos x + \cos 3 x\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos^3 x\) | \(=\) | \(\ds \dfrac {3 \cos x + \cos 3 x} 4\) | dividing both sides by $4$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.56$