设a=arctan(1/4),则：sina=1/√5,cosa=4/√5
f(x)=2sinx-cos^2(x/2)
=4sin(x/2)cos(x/2)-cos^2(x/2)
=(4sin(x/2)-cos(x/2))cos(x/2)
=√5(sin(x/2)cosa-cos(x/2)sina)cos(x/2)
=√5sin(x/2-a)cos(x/2)
f(x)=0
即cos(x/2)=0,得α=π
或 sin(x/2-a)=0,得β=2a
sin(α+β)=sin(π+2a)=-sin(2a)=-sinacosa=-4/5