A geometric explanation is very intutive.
Look at sin(x) curve below. Sin(-1) = -Sin(1). it is not symmetric
But look at the cos(x) curve below. cos(-1) = cos(1). it is symmetric
We can write cos (-x) = cos (0-x), such as the following identity may now be used:
cos (a-b) = cos a*cos b + sin a*sin b
Let a = 0 and b = x
cos (0-x) = cos 0*cos x + sin 0*sin x
But cos 0 = 1 and sin 0 = 0.
cos (0 - x) = 1*cos x + 0*sin x
cos (0 - x) = cos x
Therefore, the identity cos(-x) = cos x is verified.