Even and Odd Graphs

The identities show in the graphs. Cosine is even, so its graph is the same on both sides of the y-axis: fold the page along the axis and the halves match. That is what $\cos\left(-x\right) = \cos\left(x\right)$ says: the height at -x is the height at x.

y = cos x: a mirror image across the y-axis.

Sine is odd. Its height at -x is minus its height at x, so the left half is the right half turned upside down: a half turn about the origin carries the graph onto itself. Drawing $y = \cos\left(-x\right)$ therefore gives the cosine wave again, unchanged.