f(x)=4sinx*sin^2(π/4+x/2)+cos2x
=4sinx(1-cos(π/2+x))/2+cos2x
=2sinx(1-sinx)+cos2x
=2sinx-2sin^2x+1-2sin^2x
=-4sin^2x+2sinx+1
π/6≤x≤2/3π 1/2