Euler's Cosine Identity/Real Domain/Proof 2

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:

\(\ds \frac {e^{i x} + e^{-i x} } 2\)

\(=\)

\(\ds \frac {\cos x + i \sin x + \map \cos {-x} + i \map \sin {-x} } 2\)

\(\ds \)

\(=\)

\(\ds \frac {\cos x + \map \cos {-x} } 2\)

Sine Function is Odd

\(\ds \)

\(=\)

\(\ds \frac {2 \cos x} 2\)

Cosine Function is Even

\(\ds \)

\(=\)

\(\ds \cos x\)

$\blacksquare$