<p>The value of <span>\((\cos 65° + \sqrt{3}\sin 5° + \cos 5°)^2 = \lambda \cos^2 25°\)</span>; find the value of <span>\(\lambda\)</span></p>
Step-by-Step Solution
Key Concept: Recognize that 65° = 90° - 25° and 5° = 30° - 25°, then use complementary angle formulas and angle subtraction identities to simplify the expression into a form involving cos 25°.
<p><strong>Step 1: Rewrite angles in terms of 25°</strong></p><p>Note that 65° = 90° - 25° and 5° = 30° - 25°.</p><p>So cos 65° = cos(90° - 25°) = sin 25°</p><p><strong>Step 2: Simplify √3 sin 5° + cos 5°</strong></p><p>We have √3 sin 5° + cos 5° = 2(√3/2 sin 5° + 1/2 cos 5°)</p><p>= 2(sin 5° cos 30° + cos 5° sin 30°)</p><p>= 2 sin(5° + 30°) = 2 sin 35°</p><p><strong>Step 3: Substitute back</strong></p><p>The expression becomes:</p><p>(sin 25° + 2 sin 35°)²</p><p>Since 35° = 90° - 55° and sin 35° = cos 55°, and noting that 55° = 2(25°) + 5°:</p><p>We use sin 35° = sin(60° - 25°) = sin 60° cos 25° - cos 60° sin 25°</p><p>= (√3/2) cos 25° - (1/2) sin 25°</p><p><strong>Step 4: Substitute sin 35° expression</strong></p><p>(sin 25° + 2[(√3/2) cos 25° - (1/2) sin 25°])²</p><p>= (sin 25° + √3 cos 25° - sin 25°)²</p><p>= (√3 cos 25°)²</p><p>= 3 cos² 25°</p><p><strong>Step 5: Compare with λ cos² 25°</strong></p><p>We have 3 cos² 25° = λ cos² 25°</p><p>Therefore, λ = 3</p><p><strong>∴ Answer: Q</strong></p>
Correct Answer: Q