Power Reduction Formulas/Cosine to 6th
Theorem
- $\cos^6 x = \dfrac {10 + 15 \cos 2 x + 6 \cos 4 x + \cos 6 x} {32}$
where $\cos$ denotes cosine.
Proof
| \(\ds \cos 6 x\) | \(=\) | \(\ds 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x - 1\) | Sextuple Angle Formula for Cosine | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 32 \cos^6 x\) | \(=\) | \(\ds \cos 6 x + 48 \cos^4 x - 18 \cos^2 x + 1\) | rearranging | ||||||||||
| \(\ds \) | \(=\) | \(\ds \cos 6 x + 48 \paren {\dfrac {3 + 4 \cos 2 x + \cos 4 x} 8} - 18 \paren {\dfrac {1 + \cos 2 x} 2} + 1\) | Fourth Power of Cosine, Square of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos 6 x + \paren {18 + 24 \cos 2 x + 6 \cos 4 x} - \paren {9 + 9 \cos 2 x} + 1\) | multipying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds 10 + 15 \cos 2 x + 6 \cos 4 x + \cos 6 x\) | rearranging | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cos^6 x\) | \(=\) | \(\ds \frac {10 + 15 \cos 2 x + 6 \cos 4 x + \cos 6 x} {32}\) | dividing both sides by $32$ |
$\blacksquare$