Solution:

$\begin{aligned} \sin 420^{\circ} &=\sin \left(90^{\circ} \times 4+60^{\circ}\right) \\ &=\sin 60^{\circ}=\frac{\sqrt{3}}{2} \\ \cos 390^{\circ} &=\cos \left(90^{\circ} \times 4+30^{\circ}\right) \\=& \cos 30^{\circ}=\frac{\sqrt{3}}{2} \\ &=\cos \left(90^{\circ} \times 3+30^{\circ}\right) \\ &=\sin 30^{\circ}=\frac{1}{2} \end{aligned}$

$\begin{aligned} \sin \left(-330^{\circ}\right) &=-\sin \left(330^{\circ}\right) \\ &=-\sin \left(90^{\circ} \times 3+60^{\circ}\right) \\ &=-\left(-\cos 60^{\circ}\right)=\frac{1}{2} \\ \sin 420^{\circ} \cos 390^{\circ}+\cos \left(-300^{\circ}\right) \sin \left(-330^{\circ}\right) \\=\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)+\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) \\=1 \end{aligned}$