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

\(\displaystyle \cos 3 x\) \(=\) \(\displaystyle 4 \cos^3 x - 3 \cos x\) Triple Angle Formula for Cosine
\(\displaystyle \leadsto \ \ \) \(\displaystyle 4 \cos^3 x\) \(=\) \(\displaystyle 3 \cos x + \cos 3 x\) rearranging
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cos^3 x\) \(=\) \(\displaystyle \dfrac {3 \cos x + \cos 3 x} 4\) dividing both sides by $4$

$\blacksquare$


Sources