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Question:

The sum of all values of θ∈(0,π/2) satisfying sin²2θ + cos⁴2θ = 3/4 is?

5π/4

π/2

π

3π/8

Solution:

sin²2θ + cos⁴2θ = 3/4, θ ∈ (0, π/2) ⇒ 1 - cos²2θ + cos⁴2θ = 3/4 ⇒ 4cos⁴2θ - 4cos²2θ + 1 = 0 ⇒ (2cos²2θ - 1)² = 0 ⇒ cos²2θ = 1/2 = cos²(π/4) ⇒ 2θ = nπ ± π/4, n ∈ I ⇒ θ = nπ/2 ± π/8 ⇒ θ = π/8, π/2 - π/8Sum of solutions π/2.