Angle sum identity:
[tex]\sin285^\circ=\sin(240^\circ+45^\circ)=\sin240^\circ\cos45^\circ+\cos240^\circ\sin45^\circ[/tex]
Now
[tex]\sin240^\circ=\sin(180^\circ+60^\circ)=-\sin60^\circ=-\dfrac{\sqrt3}2[/tex]
[tex]\cos240^\circ=\cos(180^\circ+60^\circ)=-\cos60^\circ=-\dfrac12[/tex]
[tex]\sin45^\circ=\cos45^\circ=\dfrac1{\sqrt2}[/tex]
so we end up with
[tex]\sin285^\circ=-\dfrac{\sqrt3}{2\sqrt2}-\dfrac1{2\sqrt2}=-\dfrac{\sqrt3+1}{2\sqrt2}=-\dfrac{\sqrt6+\sqrt2}4[/tex]
