If X = tan(A + B), then using the addition formula for tangent, which of the following expressions correctly represents X in terms of tan A and tan B?

Introduction / Context:
This question directly tests your knowledge of the tangent addition formula in trigonometry. Knowing how to express tan(A + B) in terms of tan A and tan B is essential for simplifying expressions and solving trigonometric equations. Recognising the correct formula from among several similar looking options is a common test of attention to detail in exams.

Given Data / Assumptions:

• X is defined as tan(A + B).
• We are asked to express X in terms of tan A and tan B.
• A and B are angles for which all functions involved are defined.

Concept / Approach:
The standard tangent addition identity is: tan(A + B) = (tan A + tan B) / (1 - tan A tan B). We simply recall this identity and match it with one of the given options. Other options involve sign changes or extra terms that correspond to different, incorrect formulas. Being able to distinguish the correct formula from minor variations is crucial.

Step-by-Step Solution:
Recall the exact identity: tan(A + B) = (tan A + tan B) / (1 - tan A tan B). Since X = tan(A + B), we have X = (tan A + tan B) / (1 - tan A tan B). Now compare this with the options: Option A: (tan A - tan B) / (1 + tan A tan B) has wrong signs in both numerator and denominator. Option B: (tan A + tan B) / (1 - tan A tan B) matches the identity exactly. Option C: (tan A + tan B) / (1 + tan A tan B) has a plus in the denominator, which is incorrect for tan(A + B). Option D: (tan A - tan B) / (1 - tan A tan B) corresponds to tan(A - B), not tan(A + B). Option E adds +1 at the end and does not match any standard formula. Therefore, option B is the correct expression for X.

Verification / Alternative check:
As a quick check, pick A = B = 45°. Then A + B = 90°, so tan(A + B) is undefined (infinite). In our formula, tan 45° = 1, so the denominator 1 - tan A tan B becomes 1 - 1 * 1 = 0, while the numerator is 1 + 1 = 2, giving division by zero, which matches the undefined nature of tan 90°. The other formulas would not all show this behaviour consistently.

Why Other Options Are Wrong:
Option A corresponds to a different combination of angles. Option C would be the form for something like tan(A + B) if the identity had a plus in the denominator, which it does not. Option D is actually the formula for tan(A - B). Option E modifies the correct formula by adding 1, which changes the functional behaviour entirely and does not represent a standard trigonometric identity.

Common Pitfalls:
Students often mix up tan(A + B) and tan(A - B) or misremember the sign in the denominator. Another frequent mistake is to flip the numerator or denominator when trying to recall the formula from memory. Practising these identities regularly and checking them with simple angles helps prevent such errors.

Final Answer:
The correct expression for X is (tan A + tan B) / (1 - tan A tan B).