Verified answer

sin(a-2π/3)-sin(a+2π/3)=2sin((a-2π/3-a-2π/3)/2)*cos((a-2π/3+a+2π/3)/2))=

=2sin(-4π/6)*sin(2a/2)= -2sin(2π/3)*cosa=-2*sqrt{3}/2 *cosa=-sqrt{3}*cosa