<p>Find the value of \(16(\sin^2 18^\circ + \sin^2 36^\circ + \sin^2 54^\circ + \sin^2 72^\circ)\).</p>
Step-by-Step Solution
Key Concept: Use complementary angle relationships (sin(90°-θ) = cos θ) to pair terms, then apply the identity sin²θ + cos²θ = 1 strategically across the four angles.
<p><strong>Step 1:</strong> Recognize complementary angle pairs.</p><p>Note that 54° = 90° - 36° and 72° = 90° - 18°</p><p>Therefore: sin 54° = cos 36° and sin 72° = 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.</p><p>= (sin²18° + cos²18°) + (sin²36° + cos²36°)</p><p><strong>Step 4:</strong> Apply the Pythagorean identity sin²θ + cos²θ = 1.</p><p>= 1 + 1 = 2</p><p><strong>Step 5:</strong> Multiply by 16.</p><p>16 × 2 = <strong>32</strong></p><p>∴ Answer: <strong>32</strong></p>
Correct Answer: 32