m+n=a(cos3θ+3cos2θsinθ+3cosθsin2θ+sin3θ)=a(cosθ+sinθ)3;
cosθ+sinθ=∛[(m+n)/a];
similarly m-n=a(cosθ-sinθ)3; cosθ-sinθ=∛[(m-n)/a].
(cosθ+sinθ)2+(cosθ-sinθ)2=2cos2θ+2sinθcosθ-2sinθcosθ+2sin2θ=2.
Therefore (∛[(m+n)/a])2+(∛[(m-n)/a])2=2,
multiply through by a⅔: (m+n)⅔+(m-n)⅔=2a⅔ QED