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.