Cosine Function is Even/Proof 2

Theorem

$\map \cos {-z} = \cos z$

That is, the cosine function is even.

Proof

\(\ds \map \cos {-z}\)

\(=\)

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

Euler's Cosine Identity

\(\ds \)

\(=\)

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

simplifying

\(\ds \)

\(=\)

\(\ds \cos z\)

Euler's Cosine Identity

$\blacksquare$