I believe (sinA + sinB)(sinA - sinB) = [2sin(A+B)/2*(sinA + sinB)(sinA - sinB) = [2sin(A+B)/2*cos(A+B)/2][2sin(A-B)/2*cos(A-B)/2]cos(A-B)/2][2sin(A-B)/2*cos(A+B)/2]

and further... how do we get from [2sin(A+B)/2*cos(A+B)/2][2sin(A-B)/2*cos(A-B)/2] to sin(A+B)sin(A-B)