Indentity

sin(A+B)sin(A--B)
= (sinAcosB+cosAsinB)(sinAcosB--cosAsinB)
= (sinAcosB)^2 -- (cosAsinB)^2
= sin^2A(1--sin^2B) --sin^2B(1--sin^2A)
= sin^2A -- sin^2Asin^2B -- sin^2B + sin^2Asin^2B
= sin^2A -- sin^2B

sin²A-sin²B
=(sinA-sinB)(sinA+sinB)
=2sin(A+B2cos(A+B2)*2sin(A-B2)cos(A-B2)
=sin(A+B)sin(A-B)

Also,

sin(A+B).sin(A-B)

= 1/2[cos2B - cos2A]

=1/2[1 - 2sin²B - 1 + 2sin²A]

= sin²A - sin²B.