Power Reduction Formulas/Cosine to 5th

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Theorem

$\cos^5 x = \dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$

where $\cos$ denotes cosine.


Proof

\(\displaystyle \cos 5 x\) \(=\) \(\displaystyle 16 \cos^5 x - 20 \cos^3 x + 5 \cos x\) Quintuple Angle Formula for Cosine
\(\displaystyle \leadsto \ \ \) \(\displaystyle 16 \cos^5 x\) \(=\) \(\displaystyle \cos 5 x + 20 \cos^3 x - 5 \cos x\) rearranging
\(\displaystyle \) \(=\) \(\displaystyle \cos 5 x + 20 \paren {\frac {3 \cos x + \cos 3 x} 4} - 5 \cos x\) Power Reduction Formula for Cube of Sine
\(\displaystyle \) \(=\) \(\displaystyle \cos 5 x + 15 \cos x + 5 \cos 3 x - 5 \cos x\) multipying out
\(\displaystyle \) \(=\) \(\displaystyle 10 \cos x + 5 \cos 3 x + \cos 5 x\) rearranging
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cos^5 x\) \(=\) \(\displaystyle \frac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}\) dividing both sides by 16

$\blacksquare$


Sources