<p>The period of the function \(f(x) = \sin^4 x + \cos^4 x\) is:</p>
Step-by-Step Solution
Key Concept: Use the algebraic identity (a² + b²)² = a⁴ + b⁴ + 2a²b², combined with sin²x + cos²x = 1 and the double angle formula cos(2x) = cos²x - sin²x to reduce to a function involving cos(2x), whose period is π/2.
<p><strong>Step 1:</strong> Simplify f(x) = sin⁴x + cos⁴x using the identity a⁴ + b⁴ = (a² + b²)² - 2a²b²</p><p>sin⁴x + cos⁴x = (sin²x + cos²x)² - 2sin²x·cos²x = 1 - 2sin²x·cos²x</p><p><strong>Step 2:</strong> Express sin²x·cos²x in terms of sin(2x). Since sin(2x) = 2sinx·cosx, we have sin²x·cos²x = sin²(2x)/4</p><p>f(x) = 1 - 2·(sin²(2x)/4) = 1 - sin²(2x)/2</p><p><strong>Step 3:</strong> Use sin²(2x) = (1 - cos(4x))/2</p><p>f(x) = 1 - (1 - cos(4x))/4 = 1 - 1/4 + cos(4x)/4 = 3/4 + cos(4x)/4</p><p><strong>Step 4:</strong> The period of cos(4x) is 2π/4 = π/2</p><p>∴ Answer: B (Period = π/2)</p>
Correct Answer: B