If u=a2cos2θ+b2sin2θ+a2sin2θ+b2cos2θ then the difference between the maximum and minimum of u2 is given by

u2=a2+b2+2(a2cos2θ+b2sin2θ)(a2sin2θ+b2cos2θ)=a2+b2+2(a4+b4)sin2θcos2θ+a2b2(cos4θ+sin4θ)=a2+b2+2(a4+b4)sin2θcos2θ+a2b2(1−2sin2θcos2θ)=a2+b2+2a2b2+(a2−b2)2sin2θcos2θ=a2+b2+2a2b2+(a2−b2)24sin22θ∴difference =(maximum of u2)-(minimum of u2)=2a2b2+(a2−b2)4−2ab=(a2+b2)2−2ab=(a−b)2