<p>If \(\dfrac{1}{16}(\cos 36^\circ \sin 54^\circ)^2 - \left(\dfrac{1}{4}\sin 36^\circ \sin 36^\circ\right)^2 \equiv \dfrac{\sqrt{a}-b}{c}\), find \((a + b + c)\).</p>
Step-by-Step Solution
Key Concept: Recognize that sin 54° = cos 36° (complementary angles), then use the identity cos²36° - sin²36° = cos 72° along with the exact value cos 36° = (√5 + 1)/4 to evaluate the expression.
<p><strong>Step 1:</strong> Simplify using complementary angle: sin 54° = cos 36°</p><p>Expression becomes: (1/16)(cos 36° · cos 36°)² - (1/16)(sin 36°)²</p><p>= (1/16)[cos⁴36° - sin⁴36°]</p><p><strong>Step 2:</strong> Factor difference of squares: cos⁴36° - sin⁴36° = (cos²36° - sin²36°)(cos²36° + sin²36°)</p><p>= (cos²36° - sin²36°)(1) = cos 72°</p><p><strong>Step 3:</strong> Use exact value cos 72° = (√5 - 1)/4</p><p>Expression = (1/16) · (√5 - 1)/4 = (√5 - 1)/64</p><p><strong>Step 4:</strong> Match with (√a - b)/c form:</p><p>√a = √5, so a = 5</p><p>b = 1, c = 64</p><p>∴ a + b + c = 5 + 1 + 64 = <strong>70</strong></p>
Correct Answer: 70