Cosine Exponential Formulation/Real Domain/Proof 2

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Theorem

$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$


Proof

Recall Euler's Formula:

$e^{i x} = \cos x + i \sin x$


Then, starting from the right hand side:

\(\displaystyle \frac {e^{i x} + e^{-i x} } 2\) \(=\) \(\displaystyle \frac {\cos x + i \sin x + \cos \paren {-x} + i \sin \paren {-x} } 2\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\cos x + \cos \paren {-x} } 2\) Sine Function is Odd
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 \cos x} 2\) Cosine Function is Even
\(\displaystyle \) \(=\) \(\displaystyle \cos x\)

$\blacksquare$