Question

Is the function `f(x) = cos x` even, odd or neither?

Answer 1

Even

Explanation:

`cos(x)=cos(-x)`, therefore cosine is an even function.

Read more:
http://mathworld.wolfram.com/Cosine.html

Answer 2

To prove that `cos(theta)` is even, i.e. that `cos(-theta)=cos(theta)`, we can use the unit circle, which mind you, is the definition of cosine arguments outside the interval `[0,pi/2]`.

The unit circle is a circle of radius one centered at the origin. We can draw the following constructions for `cos(theta)` and `cos(-theta)`:

We see that the points `(cos(theta),sin(theta))` and `(cos(-theta),sin(-theta))` are on the same vertical line. Since the unit circle is in a cartesian coordinate system, this must mean they have the same `x`-coordinates.

The first point has an `x`-coordinate of `cos(theta)`, and the second has an `x`-coordinate of `cos(-theta)`, and they must be equal, so it quite easily follows that:
`cos(-theta)=cos(theta)`

Which proves that cosine is an even function.