[tex]we-have:cos2x+cos2y+2sin(x+y)=2\Leftrightarrow 1-2sin^2x+1-2sin^2y+2sin(x+y)=2\Leftrightarrow sin^2x+sin^2y-sinx.cosy-cosx.siny=0\Leftrightarrow sinx(sinx-cosy)+siny(siny-cosx)=0\Leftrightarrow sinx.\frac{sin^2x-cos^2y}{sinx+cosy}+siny.\frac{sin^2y-cos^2x}{siny+cosx}=0\Leftrightarrow sinx.\frac{1-cos^2x-cos^2y}{sinx+cosy}+siny.\frac{1-cos^2y-cos^2x}{siny+cosx}=0\Rightarrow 1-cos^2x-cos^2y=0(because:x;y\epsilon (0;\frac{\Pi }{2}))\Rightarrow cos^2x=1-cos^2y\Leftrightarrow cos^2x=sin^2y\Rightarrow cosx=siny\Rightarrow x+y=\frac{\Pi }{2}[/tex]
[tex]So:\frac{sin^4x}{y}+\frac{sin^4y}{x}\geqslant \frac{(cos^2y+sin^2y)^2}{y+x}=\frac{1}{\frac{\Pi }{2}}=\frac{2}{\Pi }[/tex]