Power Reduction Formulas/Cosine Squared/Proof 2

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Theorem

$\cos^2 x = \dfrac {1 + \cos 2 x} 2$


Proof

\(\displaystyle \dfrac {1 + \cos 2 x} 2\) \(=\) \(\displaystyle \dfrac 1 2 \left({1 + \dfrac {e^{2 i x} + e^{-2 i x} } 2}\right)\) Cosine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 4 \left({e^{2 i x} + 2 + e^{-2 i x} }\right)\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 4 \left({e^{2 i x} + 2 \left({e^{i x} }\right) \left({e^{-i x} }\right) + e^{-2 i x} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\dfrac {e^{i x} + e^{-i x} } 2}\right)^2\) Square of Sum
\(\displaystyle \) \(=\) \(\displaystyle \cos^2 x\) Cosine Exponential Formulation

$\blacksquare$