Euler's Cosine Identity/Proof 2

Theorem

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

Proof

Recall Euler's Formula:

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

Then, starting from the right hand side:

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

\(=\)

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

\(\ds \)

\(=\)

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

Sine Function is Odd

\(\ds \)

\(=\)

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

Cosine Function is Even

\(\ds \)

\(=\)

\(\ds \cos z\)

$\blacksquare$