The sum of all real solutions of sin^6 x + cos^6 x = 1/4 in [0, 2π] is:
Step-by-Step Solution
Key Concept: Use the identity sin^6 x + cos^6 x = 1 - 3 sin^2 x cos^2 x.
Step 1: sin^6 x + cos^6 x = 1 - 3 sin^2 x cos^2 x = 1/4. Step 2: 3 sin^2 x cos^2 x = 3/4 => sin^2 x cos^2 x = 1/4 => (1/4) sin^2 2x = 1/4 => sin^2 2x = 1 => sin 2x = ±1. Step 3: 2x = π/2 + nπ => x = π/4 + nπ/2. In [0, 2π], solutions are π/4, 3π/4, 5π/4, 7π/4. Step 4: Sum = (1+3+5+7)π/4 = 16π/4 = 4π.
Correct Answer: C