Think of the unit circle. You'll see that cos(x + pi) = - cos(x). So cos(x) - cos(x + pi) = cos(x) - (-cos(x)) = 2 cos(x). Then divide that by 2.
If you draw the unit circle and label the quadrants with which trig values are positive it is easy to visualise.
You can see that adding 180 degrees onto an angle brings a point on the circle from a positive quadrant to a negative quadrant or vice versa. But when you add 180 degrees to a point it remains the same angle from the horizontal axis.
Taking these two factors into account you can see that cos(x)= -cos(x+180)