<p><strong>811.</strong> In \(\triangle ABC\), if \(\sin A \sin B \sin C + \cos A \cos B = 1\) then the value of \(\cos^2 A + \sin^2 B + 2\sin^2 \dfrac{C}{2}\) is:</p>
Step-by-Step Solution
Key Concept: Use the constraint sin A sin B sin C + cos A cos B = 1 to deduce relationships between angles. Since sin A sin B sin C ≤ 1 and cos A cos B ≤ 1, equality forces specific angle values; apply cos(A+B) = cos C in a triangle.
<p><strong>Step 1:</strong> Analyze the constraint: sin A sin B sin C + cos A cos B = 1</p><p>Since in a triangle A + B + C = π, we have C = π - (A + B), so sin C = sin(A + B) and cos C = -cos(A + B).</p><p><strong>Step 2:</strong> Rewrite the constraint using sin(A + B) = sin A cos B + cos A sin B:</p><p>sin A sin B · sin(A + B) + cos A cos B = 1</p><p>sin A sin B(sin A cos B + cos A sin B) + cos A cos B = 1</p><p>sin²A sin B cos B + sin A sin B cos A sin B + cos A cos B = 1</p><p><strong>Step 3:</strong> Rearrange: sin²A sin B cos B + sin²A sin²B cos A + cos A cos B = 1</p><p>cos A cos B(1 + sin²A sin B) + sin²A sin B cos B = 1</p><p><strong>Step 4:</strong> For this equation to hold with the constraint that both terms are bounded, test if A = B and C = π/2:</p><p>• sin A sin A sin(π/2) + cos A cos A = sin²A + cos²A = 1 ✓</p><p><strong>Step 5:</strong> When C = π/2 and A = B, we have A = B = π/4.</p><p><strong>Step 6:</strong> Calculate cos²A + sin²B + 2sin²(C/2):</p><p>cos²(π/4) + sin²(π/4) + 2sin²(π/4)</p><p>= (1/√2)² + (1/√2)² + 2(1/√2)²</p><p>= 1/2 + 1/2 + 2(1/2)</p><p>= 1/2 + 1/2 + 1</p><p>= 2</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: 2