$A$, $B$ and $C$ are the angles of a triangle. Given that $\sin(A + B) = \frac{12}{13}$.

1. Relate angle $C$ to $A+B$ using the property of a triangle.

Since $A$, $B$, and $C$ are angles of a triangle, $A + B + C = 180^\circ$.
Therefore, $C = 180^\circ - (A + B)$.

2. Find $\sin C$ using the supplementary angle identity.

$\sin C = \sin(180^\circ - (A + B)) = \sin(A + B) = \frac{12}{13}$

3. Determine the possible values of $\tan C$.

Since $\sin C = \frac{12}{13}$ and $C$ is an angle in a triangle, $C$ can be acute or obtuse.
Using the Pythagorean identity $\sin^2 C + \cos^2 C = 1$:
$\cos^2 C = 1 - \left(\frac{12}{13}\right)^2 = \frac{25}{169}$
$\cos C = \pm \frac{5}{13}$

Thus, $\tan C = \frac{\sin C}{\cos C} = \frac{12/13}{\pm 5/13} = \pm \frac{12}{5}$