If x is defined by the expression: x = (tan A + tan B) / (1 − tan A tan B), then x is equal to which standard trigonometric expression (assuming all terms are defined)?

Introduction / Context:
This question tests recognition of a standard trigonometric identity. Many aptitude and simplification questions become easy if you can identify an expression as tan(A + B), tan(A - B), or related forms. The given form with (tan A + tan B) in the numerator and (1 - tan A tan B) in the denominator is the classic tangent addition identity. The main task is to match it correctly and not confuse it with the subtraction identity, which has a sign change in the numerator and denominator.

Given Data / Assumptions:

• x = (tan A + tan B) / (1 - tan A tan B) • Assume denominators are non-zero and tan values are defined • Required: identify x as a known trig expression

Concept / Approach:
Use the identity: tan(A + B) = (tan A + tan B) / (1 - tan A tan B). This identity is derived from sin(A + B)/cos(A + B) using the sine and cosine addition formulas, but for this question, recognition is sufficient. Compare the structure and signs carefully to ensure it matches the addition form and not the subtraction form: tan(A - B) = (tan A - tan B) / (1 + tan A tan B).

Step-by-Step Solution:
1) Write the standard identity for tan(A + B): tan(A + B) = (tan A + tan B) / (1 - tan A tan B) 2) Compare with the given expression for x: x = (tan A + tan B) / (1 - tan A tan B) 3) The numerator and denominator match exactly (including signs). 4) Therefore: x = tan(A + B)

Verification / Alternative check:
As a quick check, take A = 30° and B = 15°: tan 30° ≈ 0.577 and tan 15° ≈ 0.268. Compute the expression: (0.577 + 0.268)/(1 - 0.577*0.268) ≈ 0.845/(1 - 0.155) ≈ 0.845/0.845 ≈ 1. tan(45°) = 1, and 30° + 15° = 45°, confirming the identity match.

Why Other Options Are Wrong:
• tan(A - B) uses (tan A - tan B) and (1 + tan A tan B), so the signs do not match. • cot(A ± B) would be the reciprocal of tan(A ± B), not the same expression. • sec(A + B) is unrelated to this ratio structure.

Common Pitfalls:
• Confusing the denominator sign: addition has (1 - tan A tan B), subtraction has (1 + tan A tan B). • Thinking cot(A + B) is the same because it is “related,” but it is actually the reciprocal.

Final Answer:
tan(A + B)