1. **State the problem:** Evaluate the expression $\sin(420^\circ) \cos(390^\circ) + \cos(-300^\circ) \sin(-330^\circ)$.\n\n2. **Recall the formula:** This expression resembles the sine addition formula: $$\sin(A+B) = \sin A \cos B + \cos A \sin B.$$\n\n3. **Identify angles:** Let $A = 420^\circ$ and $B = -330^\circ$. Then the expression matches $$\sin A \cos B + \cos A \sin B = \sin(A+B).$$\n\n4. **Calculate $A+B$:** $$420^\circ + (-330^\circ) = 90^\circ.$$\n\n5. **Evaluate $\sin(90^\circ)$:** $$\sin(90^\circ) = 1.$$\n\n6. **Final answer:** The value of the expression is $1$.