What is the relationship in the tangent formula tan[x(+/-)y]?

The tangent formula for the addition and subtraction of angles states that:

[

\tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}

]

This formula shows that when adding or subtracting two angles, the numerator reflects the relationship between the two tangents, using the addition or subtraction (depending on if you are looking at (x + y) or (x - y)). The signs in the numerator correspond to whether you are using addition or subtraction, which is crucial for correctly applying the formula.

In the context of the provided question, saying that the top and bottom are opposite of each other accurately describes how the signs in the numerator change depending on the operation (addition or subtraction). This relationship is critical to achieving the correct result when calculating the tangent of an angle that is a sum or difference of two other angles since it ensures the correct interpretation of the relationship between the angles involved.

Thus, the correct choice about the relationship indicates that the signs in the numerator will switch according to whether you are adding or subtracting (x) and (y).