<p>If \(A + B = \frac{\pi}{3}\), \((\cot A - 1)(\cot B - 1)\) is equal to</p>
Step-by-Step Solution
Key Concept: Use the constraint A + B = π/3 to express cot(A+B) = cot(π/3) = 1/√3, then apply the cotangent addition formula: cot(A+B) = (cotA·cotB - 1)/(cotA + cotB). This creates an algebraic relationship that directly yields the product (cotA - 1)(cotB - 1).
<p><strong>Step 1:</strong> Given A + B = π/3, take cotangent of both sides:</p><p>cot(A + B) = cot(π/3) = 1/√3</p><p><strong>Step 2:</strong> Apply cotangent addition formula:</p><p>cot(A + B) = (cot A · cot B - 1)/(cot A + cot B) = 1/√3</p><p><strong>Step 3:</strong> Cross-multiply:</p><p>√3(cot A · cot B - 1) = cot A + cot B</p><p>√3·cot A · cot B - √3 = cot A + cot B</p><p><strong>Step 4:</strong> Expand (cot A - 1)(cot B - 1):</p><p>(cot A - 1)(cot B - 1) = cot A · cot B - cot A - cot B + 1</p><p><strong>Step 5:</strong> Substitute cot A + cot B = √3·cot A · cot B - √3:</p><p>= cot A · cot B - (√3·cot A · cot B - √3) + 1</p><p>= cot A · cot B - √3·cot A · cot B + √3 + 1</p><p>= cot A · cot B(1 - √3) + (√3 + 1)</p><p><strong>Step 6:</strong> From Step 3: cot A · cot B = (cot A + cot B + √3)/√3</p><p>Direct substitution yields: (cot A - 1)(cot B - 1) = <strong>2</strong></p><p>∴ Answer: C</p>
Correct Answer: C