Answer
$-\frac{\sqrt3}{2}.$
Work Step by Step
Recall that
$\sin(\alpha-\beta)=\sin(\alpha)\cos(\beta)-\cos(\alpha)\sin(\beta)$.
Hence,
$\sin(20^\circ)\cos(80^\circ)+\cos(80^\circ)\sin(20^\circ)=\sin(20^\circ-80^\circ)=\sin(-60^\circ)=-\frac{\sqrt3}{2}.$
