How to prove this trig identity

cos(-x) + cos(pi - x) = cos(pi + x) + cosx

Well cos(-x) = cos(x) and as sin(π) = 0

cos(π - x) = cos(π)cos(x) + sin(π)sin(x) = cos(π)cos(x) - sin(π)sin(x) = cos(π - x).

Hence cos(-x) + cos(π- x) = cos(x) + cos(π - x) = cos(π + x) + cos(x).