Prove cos^3 x = cos x

This is only true for selected values of x

cos^3x = cosx

cos^3x - cosx = 0

cosx(cos^2x - 1)  = 0

cosx(cosx - 1)(cosx + 1) = 0

So, either cosx = 0    and this is true at x = pi/2 ± (pi/2)n  for some integer n

Or

cosx - 1 = 0

cos x = 1    and this is true at x = 0 ± (2pi)n  for some integer n

Or

cosx + 1 = 0

cosx = -1   and this is true at x = pi ± (2pi)n   for some integer n