We can use our knowledge of additional formulae...
cos(A+B) = cosAcosB - sinAsinB
cos^2 (x+pi/3) = ( cosxcos(pi/3) - sinx sin(pi/3) )^2
= (1/2cosx - sqrt(3)/2 sinx )^2 = 1/4cos^2x -sqrt(3)/2 sinxcosx +3/4 sin^2 x
cos^2 (x-pi/3) = ( cosxcos(pi/3) + sinxsin(pi/3) )^2
= (1/2cosx + sqrt(3)/2 sinx )^2 = 1/4cos^2x + sqrt(3)/2 sinxcosx +3/4cos^2 x
=> cos^2x + cos^2(x-pi/3) + cos^2(x+pi/3)
= cos^2x + 1/2cos^2x + 3/2 sin^2 x = 3/2cos^2x + 3/2sin^2x
-= 3/2 ( cos^2 x + sin^2 x ) = color(blue)(3/2
Using the identity sin^2 theta + cos^2 theta -= 1