For every angle A, the value of co$s^2$A + co$s^2(A+120$°) + co$s^2(A-120$°) is:

Use co$s^2$$x$ = (1 + cos2x)/2. The three constant halves add to 3/2, and the remaining half-sum is (1/2)[cos2A + cos$(2A+240$°) + cos$(2A-240$°)]. The last two combine to 2·cos2A·cos240° = −cos2A, which cancels the first, so the bracket vanishes and 3/2 remains. Check at $A=0$°: 1 + 1/4 + 1/4 = 1.5. The answer 3 would need all three cosines to be ±1 at once, which three angles 120° apart can never manage.