In trigonometric identities, express tan(A - B) in terms of tan(A) and tan(B). Select the correct formula for X where X = tan(A - B).

Introduction / Context:
This question tests your knowledge of trigonometric angle subtraction formulas, specifically the identity for tan(A − B). Being able to express tan(A − B) in terms of tan(A) and tan(B) is essential in solving many trigonometric equations and simplifying expressions, and it also illustrates the structure of tangent addition and subtraction formulas used widely in mathematics.

Given Data / Assumptions:

• A and B are angles for which tan(A) and tan(B) are defined.
• We must find the correct expression for tan(A − B) in terms of tan(A) and tan(B).
• Several candidate formulas are provided; exactly one matches the standard identity.

Concept / Approach:
The standard tangent addition and subtraction identities are:

• tan(A + B) = (tan A + tan B) / (1 − tan A * tan B)
• tan(A − B) = (tan A − tan B) / (1 + tan A * tan B)

We recall these formulas or derive them from the sine and cosine addition formulas. Then, we match the expression for tan(A − B) to the given options and select the one that agrees with the standard identity.

Step-by-Step Solution:
1) Start from the known identity: tan(A − B) = (tan A − tan B) / (1 + tan A * tan B). 2) Compare this general identity with the multiple-choice options. 3) Option a uses tan(A) + tan(B) in the numerator and 1 − tan(A)*tan(B) in the denominator, which corresponds to tan(A + B), not tan(A − B). 4) Option b still uses tan(A) + tan(B) and an incorrect denominator pattern for tan(A − B). 5) Option c has tan(A) − tan(B) in the numerator but 1 − tan(A)*tan(B) in the denominator, which does not match the subtraction identity. 6) Option d, (tan(A) − tan(B)) / (1 + tan(A)*tan(B)), matches exactly the known formula for tan(A − B). 7) Option e has a completely different structure and does not correspond to any standard addition or subtraction formula.

Verification / Alternative check:
To verify, take specific angles, such as A = 60° and B = 30°. Then tan(A − B) = tan(30°) = 1/√3. Evaluate option d: tan(60°) = √3 and tan(30°) = 1/√3, so the numerator is √3 − 1/√3 and the denominator is 1 + √3 * (1/√3) = 1 + 1 = 2. Numerically simplifying the numerator and denominator yields a result equal to 1/√3, confirming that option d matches tan(A − B). The other options give different values when evaluated with the same angles.

Why Other Options Are Wrong:
Option a is the identity for tan(A + B), not tan(A − B). Option b incorrectly mixes signs in both the numerator and denominator and does not correspond to any standard identity. Option c uses tan(A) − tan(B) but pairs it with the denominator from tan(A + B), which is mismatched. Option e does not align with any of the standard formulas and behaves very differently for typical angle values. Only option d reflects the correct formula for tan(A − B).

Common Pitfalls:
A common mistake is to confuse the signs for the addition and subtraction formulas, especially in the denominator. Some learners incorrectly assume that both numerator and denominator simply follow the sign between A and B, which is not the case for tangent. Carefully memorizing or deriving the formula from sine and cosine definitions helps avoid these sign errors and solidifies understanding.

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