Answer
$0$
Work Step by Step
Recall:
Sine is an odd function so $\sin{(-\theta)} = - \sin{\theta}$.
This means that
$\sin{(-20^{\circ})} = - \sin{20^{\circ}}$
Since $\cos{(\theta-360^{\circ})} = \cos{\theta}$, then
$\cos{380^{\circ}} = \cos{(380^{\circ}-360^{\circ})} = \cos{20^{\circ}}$
Thus,
$\dfrac{\sin{(-20^{\circ})}}{\cos{380^{\circ}}} = \dfrac{- \sin{20^{\circ}}}{\cos{20^{\circ}}}$
Recall that $\tan{\theta} =\dfrac{\sin{\theta}}{\cos{\theta}}$.
Hence,
$\dfrac{- \sin{20^{\circ}}}{\cos{20^{\circ}}} = - \tan{20^{\circ}}$
Therefore,
$\dfrac{\sin{(-20^{\circ})}}{\cos{380^{\circ}}} + \tan{20^{\circ}} = - \tan{20^{\circ}}+ \tan{20^{\circ}} =\boxed{0}$
