Answer
$-\dfrac{\sqrt 3}{2}$
Work Step by Step
Apply the difference formula:
$\sin(A-B)=\sin(A)\cos(B) -\cos(A)\sin(B)$
We know that $\sin$, $\csc$, and $\tan$ are odd trigonometric functions.
This implies that $f(-\theta)=-f(\theta)$
So, $\sin(-\theta)=-\sin(\theta)$
Therefore,
$$\sin(20^\circ) \ \cos(80^\circ) -\cos(20^\circ) \ \sin(80^\circ)\\
=\sin \ (20^\circ -80^\circ)\\
=\sin \ (-60^\circ)\\
=-\dfrac{\sqrt 3}{2}$$
