What is the value of the expression $\cos 2 A \cos 2 B+\sin ^2(A-B)-\sin ^2(A+B)$ ?

$\cos (2 A+2 B)$

cos 2A . cos 2B + sin² ( A - B ) - sin² ( A + B )

= cos 2A . cos 2B - [ sin² ( A + B ) + sin² ( A - B ) ]

{ we know,  sin² x - sin² y = sin(x + y). sin( x - y) }

= cos 2A . cos 2B - [ sin ( A + B + A - B ) .sin( A + B - A + B ) ]

= cos 2A . cos 2B -  sin 2A .sin 2B

{ using , cos ( x + y ) = cos x. cos y - sin x. sin y

= cos ( 2A + 2B )