Well, you could do a rule of three to find the angle 4π/3 in degrees. We know that π equals to 180°, so 4π/3 will be 240°.
From here on you have two simple ways I can think of. One: 240 = 180 + 60, so you could use the cos(a + b) identity.
Two: 240 = 180 + 60, so you could interpretate this as 60° after 180° degrees in the trigonometric circle. The trigonometric circle is basically a quarter of a circle that repeats itself 4 times. The value of the cossines and sines remain numeracally but change signals as it change quarters. We know that 180 is the beginning of a new quarter, so we can say that we are in the beginning of the quarter, the 0° position. So we are left with cossine(0° + 60°) after the beginning. As this quarter of the circle is negative for cossine values you would get -cossine(60)