SOLUTION: Use the complex number cosx + isinx to find the constants A, B, C in the identity cos4x = Acos^4x + Bcos^2xSin^2x + Csin^4x

z = cos(x) + i*sin(x)  ====>


 =  = de Moivre = cos(4x) + i*sin(4x)  ====>


cos^4(x) + 4*i*cos^3(x)*sin(x) + 6*i^2*cos^2(x)*sin^2(x) + 4*i^3*cos(x)*sin^3(x) + i^4*sin^4(x) = cos(4x) + i*sin(4x) ,   or, equivalently


cos^4(x) + 4*i*cos^3(x)*sin(x) - 6*cos^2(x)*sin^2(x) - 4*i*cos(x)*sin^3(x) + sin^4(x) = cos(4x) + i*sin(4x),   which implies


cos(4x) = 


which is an identity you are looking for.


So, A = 1,  B = -6,  C = 1.