Cosine Function is Even/Proof 1

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Theorem

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

That is, the cosine function is even.


Proof

Recall the definition of the cosine function:

$\displaystyle \cos z = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {z^{2 n} } {\left({2 n}\right)!} = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \cdots$


From Even Power is Non-Negative:

$\forall n \in \N: z^{2 n} = \paren {-z}^{2 n}$

The result follows.

$\blacksquare$


Sources