Simplify the LHS of tan2α−tan2β=sin(α+β)sin(α−β)cos2αcos2β
tan2α−tan2β=(tanα−tanβ)(tanα+tanβ)
=(sinαcosα−sinβcosβ)(sinαcosα+sinβcosβ)
=(sinαcosβ−cosαsinβcosαcosβ)(sinαcosβ+cosαsinβcosαcosβ)
=(sin(α−β)cosαcosβ)(sin(α+β)cosαcosβ)
=sin(α−β)sin(α+β)cos2αcos2β
=RHS