Cosine Function is Even

From ProofWiki
Jump to: navigation, search

Theorem

For all $x \in \C$:

$\cos \left({-x}\right) = \cos x$

That is, the cosine function is even.


Proof

Recall the definition of the cosine function:

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


From Even Powers are Positive, we have that:

$\forall n \in \N: x^{2n} = \left({-x}\right)^{2n}$

The result follows.

$\blacksquare$


Also see


Sources