cos(x+(pi)/3)=(cosx)/2-(sinx)sqrt(3)/2; cos^2(x+(pi)/3)=cos^2x/4-sinxcosxsqrt(3)/2+3sin^2x/4
cos(x-(pi)/3)=(cosx)/2+(sinx)sqrt(3)/2; cos^2(x-(pi)/3)=cos^2x/4+sinxcosxsqrt(3)/2+3sin^2x/4
Add these together: cos^2x/2+3sin^2x/2 and add cos^2x=(3/2)(cos^2x+sin^2x)=3/2 QED.