$4.65 \sin ^{2}\left(135^{\circ}-2 \alpha\right)-\sin ^{2}\left(210^{\circ}-2 \alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-4 \alpha\right)$.
4.65 To the first two terms, we apply formula (4.16), and to the third, $-(4.27):$

$\sin ^{2}\left(135^{\circ}-2 \alpha\right)-\sin ^{2}\left(210^{\circ}-2 \alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-4 \alpha\right)=$

$=\frac{1}{2}\left(1-\cos \left(270^{\circ}-4 \alpha\right)-1+\cos \left(420^{\circ}-4 \alpha\right)-\right.$

$\left.-\sin \left(360^{\circ}-4 \alpha\right)-\sin \left(30^{\circ}+4 \alpha\right)\right)=\frac{1}{2}\left(\sin 4 \alpha+\cos \left(60^{\circ}-4 \alpha\right)+\right.$

$\left.+\sin 4 \alpha-\sin \left(30^{\circ}+4 \alpha\right)\right)=\sin 4 \alpha$.