For the identity tan(α + β), what is the correct expression?

The expression for the tangent of the sum of two angles is derived from the unit circle and the properties of sine and cosine functions. The correct identity is given by the formula:

tan(α + β) = (tan α + tan β) / (1 - tan α tan β)

However, in the context of angles, it's important to also understand that when you add angles α and β, the overall tangent function incorporates their combined effects. The correct formula expresses how the tangent of the sum of two angles depends on the tangents of the individual angles.

To elaborate, this formula arises from the relationship between the sine and cosine definitions of tangent, where tangent is defined as the ratio of sine to cosine:

tan(α) = sin(α) / cos(α) and tan(β) = sin(β) / cos(β).

When combining these two tangents using the sine and cosine addition formulas, you end up with a single expression that reflects the interactions of the two angles. The "+" in the numerator indicates that the contributions of both tangent angles add together while the denominator represents how their interaction modifies the overall tangent value.

In the selected answer, the denominator should relate to the combinations of the two tangents in determining the tangent