If cot(A − B) is expressed in terms of cot A and cot B, which of the following formulas is correct? Assume the angles are such that all required trigonometric values are defined.

Introduction / Context:
This identity question tests whether you can transform angle-difference formulas correctly. Many students remember tan(A - B) but not cot(A - B). However, cot can be derived quickly using tan(A - B) and the reciprocal relationship cot θ = 1/tan θ, or by directly rewriting tan A and tan B in terms of cot A and cot B. The main difficulty is sign management: the difference identity has a plus sign in the denominator for tan(A - B), but after converting to cot form, the structure changes and the correct placement of (cotB - cotA) becomes essential.

Given Data / Assumptions:

• Required: cot(A - B) in terms of cot A and cot B • Use identities: tan(A - B) = (tan A - tan B) / (1 + tan A tan B) • Relationship: cot θ = 1 / tan θ • Assume all denominators are non-zero

Concept / Approach:
Start from tan(A - B) and then invert to get cot(A - B). To express everything in cot, replace tan A with 1/cot A and tan B with 1/cot B. After simplifying complex fractions by multiplying numerator and denominator by cot A cot B, you get a clean expression involving cotA cotB and a difference (cotB - cotA).

Step-by-Step Solution:
1) Start with the standard identity: tan(A - B) = (tan A - tan B) / (1 + tan A tan B) 2) Convert to cot by taking reciprocal: cot(A - B) = (1 + tan A tan B) / (tan A - tan B) 3) Replace tan A and tan B using cot: tan A = 1/cotA, tan B = 1/cotB 4) Substitute: cot(A - B) = (1 + (1/cotA)(1/cotB)) / ((1/cotA) - (1/cotB)) 5) Simplify numerator: 1 + 1/(cotA cotB) = (cotA cotB + 1)/(cotA cotB) 6) Simplify denominator: (1/cotA) - (1/cotB) = (cotB - cotA)/(cotA cotB) 7) Divide the two fractions (the cotA cotB cancels): cot(A - B) = [(cotA cotB + 1)/(cotA cotB)] / [(cotB - cotA)/(cotA cotB)] cot(A - B) = (cotA cotB + 1)/(cotB - cotA)

Verification / Alternative check:
Test with A = 60°, B = 30°: cot(A - B) = cot 30° = √3. cot 60° = 1/√3 and cot 30° = √3. Compute (cotA cotB + 1)/(cotB - cotA) = ((1/√3)*√3 + 1)/(√3 - 1/√3) = (1 + 1)/( (3 - 1)/√3 ) = 2/(2/√3) = √3. It matches exactly.

Why Other Options Are Wrong:
• Options with (cotB + cotA) mix the sign incorrectly; subtraction is essential for A - B. • Options with (cotA cotB - 1) correspond to a different identity structure and fail the numeric test. • (cotA + cotB)/(1 - cotA cotB) resembles a tangent-style structure but is not the correct cot(A - B) form.

Common Pitfalls:
• Confusing tan(A - B) with tan(A + B) and carrying the wrong sign. • Forgetting that cot(A - B) is the reciprocal of tan(A - B). • Dropping parentheses and reversing (cotB - cotA) which changes the sign.

Final Answer:
(cotA cotB + 1) / (cotB − cotA)