If sin⁴ x + cos⁴ x = 5/8, then the value of sin2x

Solution

We are given: sin⁴x + cos⁴x = 5/8 We are asked to find the value of sin(2x). We use the identity: sin⁴x + cos⁴x = (sin²x + cos²x)² - 2sin²x·cos²x But sin²x + cos²x = 1, so: sin⁴x + cos⁴x = 1 - 2sin²x·cos²x We are given: 1 - 2sin²x·cos²x = 5/8 ⇒ 2sin²x·cos²x = 1 - 5/8 = 3/8 Now, recall: sin(2x) = 2sinx·cosx So: sin²(2x) = [2sinx·cosx]² = 4sin²x·cos²x From above, we found: sin²x·cos²x = 3/16 ⇒ sin²(2x) = 4 × 3/16 = 3/4 ⇒ sin(2x) = √(3/4) = √3 / 2