For cos 60 = 1/2:

Let us assume AB= BC = CA = 2a. Therefore, BD = ½ BC = a.

Now, we know that cos angle = side adjacent to angle/hypotenuse

Therefore, in traingle ABD, cos 60 = BD/AB = a/2a = 1/2

For cos 120 = -1/2:

cos 120 = cos(180-60)

= -cos(60) (Since, cos(180-thita)= -cos(thita))

= -1/2