Final answer:
Without additional information such as the values of sin(theta) or cos(theta), we cannot definitively calculate tan(theta/2) from tan theta = -3 when cos theta < 0. The question requires a conceptual understanding rather than an exact calculation.
Step-by-step explanation:
To determine tan(theta/2) when tan theta = -3 and cos theta < 0, we must recognize that the angle theta is in the second or third quadrant (since the cosine is negative). However, since the tangent is also negative, theta must be in the second quadrant where both sine and cosine are negative. The double angle formulas, such as tan(2x) = 2tan(x) / (1 - tan^2(x)), help in these calculations, but here we are looking for halving the angle which requires a different approach.
The identity often used for halving the angle is tan(x/2) = 1 - cos(x) / sin(x), but we are not provided with sin(theta) or cos(theta) directly, nor are we asked to actually calculate the value of tan(theta/2). We are given options to choose from, suggesting a conceptual understanding is needed instead of a numerical one.
Since the actual formula or calculation of tan(theta/2) is not required, and no information supports the precise determination of tan(theta/2), the answer cannot be definitively determined from the given selections. All provided options suggest exact halving or doubling, none of which are mathematically justified without additional information.