If 2 sin⁴α + 2 cos⁴α − 1 = 0, where 0 ≤ α < π/2, then what is sin 2α + cos 2α equal to?

2(sin⁴α + cos⁴α) = 1. Using sin⁴α + cos⁴α = (sin²α + cos²α)² − 2sin²α·cos²α = 1 − (1/2)sin²(2α). So 2[1 − (1/2)sin²2α] = 1, giving 2 − sin²2α = 1, hence sin²2α = 1, so sin 2α = ±1. With 0 ≤ α < π/2, 0 ≤ 2α < π, so sin 2α = 1, meaning 2α = π/2, so cos 2α = 0. Therefore sin 2α + cos 2α = 1 + 0 = 1.