sinx=1/3-siny
故sinx-(cosy)^2=(1/3-siny)-[1-(siny)^2]
=(siny)^2-siny-2/3=(siny-1/2)^2-1/4-2/3
siny∈[-1,1],
当siny=-1时,(siny-1/2)^2取得最大值为9/4,
sinx-(cosy)^2取得最大值4/3.
当siny=1/2时,(siny-1/2)^2取得最小值为0,
sinx-(cosy)^2取得最小值-11/12.由sinx+siny=1/3得到sinx=(1/3)-siny.
故m=sinx-cos²y=[(1/3)-siny]-(1-sin²y)=[siny-(1/2)]²-(11/12).
即有m=[siny-(1/2)]²-(11/12).
由sinx+siny=1/3.且-1≤sinx≤1.可知，-2/3≤siny≤1.
故当siny=1/2时,(m)min=-11/12.
当siny=-2/3时,(m)max=4/9.