Prove that: `cos^2A+cos^2(A+pi/3)+cos^2(A-pi/3)=3/2`

Prove that : cos^(2)A+cos^(2)(A+pi/3)+cos^(2)(A-pi/3) =1/2[2cos^(2)A+2cos^(2)(A+60^(@))+2cos^(2)(A-60^(@))] =1/2[1+cos2A+1+cos(2A+120^(@))+1+cos(2A-120^(@))] 1/2[3+cos2A+cos(2A+120^(@))+cos(2A-120^(@))] 1/2[3+cos2A+2cos(2A+120^(@)+2A-120^(@))/(2).cos(2A+120^(@)-2A+120^(@))/(2)] =1/2[3+cos2A+2cos2Acos120^(@)] =1/2[3cos2A+2cos2Acos(90^(@)+30^(@))] =1/2[3+cos2A-2.cos2A.(1/2)] =1/2(3+cos2A-cos2A) =3/2 = RHS Hence Proved.

Prove that: cos^(2)A+cos^(2)(A+(pi)/(3))+cos^(2)(A-(pi)/(3))=(3)/(2)

Prove that: cos^(2)A+cos^(2)(A+(2 pi)/(3))+cos^(2)(A-(2 pi)/(3))=(3)/(2)

cos^(^^)2A+cos^(^^)2(A+2pi/3)+cos^(^^)2(A-2pi/3)=3/2 OR cos^(^^)2A+cos^(^^)2(A+pi/3)+cos^(^^)2(A-pi/3)=3/2

Prove that : cos A+cos(A+(2 pi)/(3))+cos(A+(4 pi)/(3))=0

Prove that cos^(2)x+cos^(2)(x+(pi)/(3))+cos^(2)(x-(pi)/(3))=(3)/(2)

Prove that cos^(2)x+cos^(2)(x+(pi)/(3))+cos^(2)(x-(pi)/(3))=(3)/(2)

Prove that: cos^(3)A + cos^(3)((2pi)/(3)+A) + cos^(3)((4pi)/3+A) = 3/4 cos3A

Prove that: cos^(2)pi/8 + cos^(2)(3pi)/(8) + cos^(2)(5pi)/(8)+ cos^(2)(7pi)/(8)=2

Prove that cos x+cos((2 pi)/(3)-x)+cos((2 pi)/(3)+x)=0