sin²α+sin²β=sin(α+β)=sinαcosβ+sinβcosa<=(sin²α+sin²β)^0.5(cos²α+cos²β)^0.5(柯西不等式）,即sin²α+sin²β<=cos²α+cos²β,即cos2α>=cos(π -2β),结合单调性即2α>=π -2β,即α+β>=π/2 .又因为sin²α+sin²β=sin(α+β)<=1即sinα<=sin(π/2-β)结合单调性α<=π/2-β即α+β<=π/2故只能是α+β=π/2 .毕.