<p>Find the value of \(16(\sin^2 18° + \sin^2 36° + \sin^2 54° + \sin^2 72°)\).</p>
Step-by-Step Solution
Key Concept: Use complementary angle relationships (sin 54° = cos 36°, sin 72° = cos 18°) to pair terms, then apply the identity sin²θ + cos²θ = 1 strategically.
<p><strong>Step 1:</strong> Recognize complementary angle relationships.</p><p>sin 54° = sin(90° − 36°) = cos 36°</p><p>sin 72° = sin(90° − 18°) = cos 18°</p><p><strong>Step 2:</strong> Rewrite the expression using these relationships.</p><p>sin²18° + sin²36° + sin²54° + sin²72°</p><p>= sin²18° + sin²36° + cos²36° + cos²18°</p><p><strong>Step 3:</strong> Group complementary pairs and apply sin²θ + cos²θ = 1.</p><p>= (sin²18° + cos²18°) + (sin²36° + cos²36°)</p><p>= 1 + 1 = 2</p><p><strong>Step 4:</strong> Multiply by 16.</p><p>16 × 2 = 20</p><p>∴ Answer: <strong>20</strong></p>
Correct Answer: 20