We could have just used the calculator to solve this problem, but to illustrate the concept we will use trigonometry. Let's break the problem first.
Let A = cos^-1(-12/13) = arccos(-12/13)
So the problem now is searching for the exact value of tan(A).
A is an angle whose cosine is -12/13. Also, sin(A) using the pythagorean identity can be either positive or negative sqrt(1-(-12/13)^2) = +-5/13. But the arccosine gives angles in the range of 0 deg to 180 deg, so the only possible value for sin(A) is 5/13.
Using the quotient identities, tan(A) = sin(A) / cos(A) = 5/13 / (-12/13) = -5/12.
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