<p>If \(\sin(\alpha + \beta) = 1\) and \(\sin(\alpha - \beta) = \dfrac{1}{2}\), then \(\tan(\alpha + 2\beta) \cdot \tan(2\alpha + \beta)\) is equal to:</p>
Step-by-Step Solution
Key Concept: Use the condition sin(α + β) = 1 to establish α + β = π/2, then apply sum-to-product formulas and angle addition identities to find the individual angles before computing the product.
<p><strong>Step 1:</strong> From sin(α + β) = 1, we get α + β = π/2</p><p><strong>Step 2:</strong> From sin(α - β) = 1/2, we get α - β = π/6 or 5π/6. Taking α - β = π/6 (principal value)</p><p><strong>Step 3:</strong> Solve the system:<br/>α + β = π/2<br/>α - β = π/6<br/>Adding: 2α = π/2 + π/6 = 2π/3, so α = π/3<br/>Subtracting: 2β = π/2 - π/6 = π/3, so β = π/6</p><p><strong>Step 4:</strong> Calculate α + 2β = π/3 + π/3 = 2π/3<br/>Calculate 2α + β = 2π/3 + π/6 = 5π/6</p><p><strong>Step 5:</strong> tan(2π/3) = -√3<br/>tan(5π/6) = 1/√3</p><p><strong>Step 6:</strong> tan(α + 2β)·tan(2α + β) = (-√3)·(1/√3) = -1</p><p>∴ Answer: C (which is -1)</p>
Correct Answer: C