We can simplify the expression \( \frac{\tan(310^\circ) - \tan(10^\circ)}{1 + \tan(310^\circ) \tan(10^\circ)} \) using the tangent subtraction formula. The formula states that:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Here, we can set \( A = 310^\circ \) and \( B = 10^\circ \). Thus, we can rewrite the expression as:
\[ \tan(310^\circ - 10^\circ) = \tan(300^\circ) \]
Next, we know that:
\[ \tan(300^\circ) = \tan(360^\circ - 60^\circ) = -\tan(60^\circ) \]
And since \( \tan(60^\circ) = \sqrt{3} \), we get:
\[ \tan(300^\circ) = -\sqrt{3} \]
Therefore, the simplified form of the original expression is:
\[ \frac{\tan(310^\circ) - \tan(10^\circ)}{1 + \tan(310^\circ) \tan(10^\circ)} = -\sqrt{3} \]
Thus, the final result is:
\[ \boxed{-\sqrt{3}} \]