Why is cos(pi) and cos (-pi) both equal to -1?

A full circle is ##2pi##

##pi## is half way around the circle counter-clockwise.

##-pi## is half way around the circle clockwise.

##cos(pi) = cos(-pi)## are the both ##cos## values for the same place

They are both equal to ##(-1)## because if viewed as a unit circle centered on the Cartesian origin the ##cos## is the ##x## value and, at halfway around the unit circle, ##x=-1##