$A=\sin(x+y) \times \sin(x-y)$
$A=(\sin x\times \cos y + \sin y \times \cos x)(\sin x\times \cos y - \sin y \times \cos x)$
$A=(\sin^2 x \times \cos^2 y - \sin y^2 \times \cos^2 x)$
$A=\sin^2 x (1-\sin^2 y) - \sin y^2 (1- \sin^2 x)$
$A=\sin^2 x -\sin^2 x \times \sin^2 y - \sin y^2 + \sin y^2 \times \sin^2 x$
$A=\sin^2 x -\sin^2 y$