For x>0: sinx−21sin2x=sinx(1−cosx).
x→0+limx3sinx(1−cosx)=x→0+limxsinx⋅x21−cosx=1⋅21=21.
So a=21.
For x<0 near 0: 2π(cosx+sinx)cosx→2π from below, so [2π(cosx+sinx)cosx]=1.
f(x)=b2sin(2π⋅1)=b2.
For continuity: b2=a=21.
a2+b2=41+21=43.