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 |
$\blacksquare$