<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 contain the same three sine squared terms (sin²π/7, sin²2π/7, sin²4π/7), just arranged differently. The key is to identify that these terms are identical sets, making the ratio equal to 1.
<p><strong>Step 1:</strong> Examine the numerator and denominator carefully.</p><p>Numerator: sin²(2π/7) + sin²(4π/7) + sin²(π/7)</p><p>Denominator: sin²(π/7) + sin²(2π/7) + sin²(4π/7)</p><p><strong>Step 2:</strong> Recognize that both contain exactly the same three terms: sin²(π/7), sin²(2π/7), and sin²(4π/7). Since addition is commutative, the numerator and denominator are equal.</p><p><strong>Step 3:</strong> Apply the property that any non-zero number divided by itself equals 1.</p><p>∴ Answer: <strong>1</strong> (which is option A)</p>
Correct Answer: A