<p>The value of the expression \(\dfrac{\sin^2\dfrac{2\pi}{7} + \sin^2\dfrac{4\pi}{7} + \sin^2\dfrac{\pi}{7}}{\sin^2\dfrac{\pi}{7} + \sin^2\dfrac{2\pi}{7} + \sin^2\dfrac{4\pi}{7}}\) is equal to:</p>
Step-by-Step Solution
Key Concept: Recognize that the numerator and denominator are identical expressions—just written in different orders. Since sin²(2π/7) + sin²(4π/7) + sin²(π/7) appears in both, the fraction simplifies to 1 regardless of the actual values.
<p><strong>Step 1:</strong> Identify the terms in the numerator: sin²(2π/7) + sin²(4π/7) + sin²(π/7)</p><p><strong>Step 2:</strong> Identify the terms in the denominator: sin²(π/7) + sin²(2π/7) + sin²(4π/7)</p><p><strong>Step 3:</strong> Observe that both numerator and denominator contain the same three terms: sin²(π/7), sin²(2π/7), and sin²(4π/7), just in different order.</p><p><strong>Step 4:</strong> Since addition is commutative, the numerator equals the denominator.</p><p><strong>Step 5:</strong> Therefore, the fraction equals 1.</p><p>∴ Answer: A (which is 1)</p>
Correct Answer: A