Power Reduction Formulas/Cosine to 4th/Proof 1

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Theorem

$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$


Proof

\(\displaystyle \cos^4 x\) \(=\) \(\displaystyle \left({\cos^2 x}\right)^2\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\frac {1 + \cos 2 x} 2}\right)^2\) Square of Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 + 2 \cos 2 x + \cos^2 2 x} 4\) multiplying out
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 + 2 \cos 2 x + \frac {1 + \cos 4 x} 2} 4\) Square of Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 + 4 \cos 2 x + 1 + \cos 4 x} 8\) multiplying top and bottom by $2$
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 + 4 \cos 2 x + \cos 4 x} 8\) rearrangement

$\blacksquare$