cos^4(x) + sin^4(x) + cos(x - π/4).sin(3x - π/4)
- 3/2 = 0
<=> 1 - 2sin^2(x).cos^2(x) + 1/2.sin2x
+ 1/2.sin(4x - π/2) - 3/2 = 0
<=> - 1/2.sin^2(2x) + 1/2.sin2x + 1/2.cos4x -1/2=0
<=> - 1/2.sin^2(2x) + 1/2sin2x + 1/2.(1-2sin^2(2x)) - 1/2 = 0
<=> -3/2.sin^2(2x) + 1/2.sin2x = 0
<=> sin2x(-3/2.sin2x + 1/2) = 0
<=> sin2x = 0 hay sin2x = 1/3
<=> 2x = kπ hay 2x = arcsin(1/3) + k2π hay 2x = π - arcsin(1/3) + k2π
<=> x = kπ/2 hay x = 1/2.arcsin(1/3) + kπ hay x = π/2 - 1/2.arcsin(1/3) + kπ
Ta có: 0 ≤ x ≤ 2π
<=> 0 ≤ kπ/2 ≤ 2π hay 0 ≤ 1/2.arcsin(1/3) + kπ ≤ 2π hay 0 ≤ π/2 - 1/2.arcsin(1/3) + kπ ≤ 2π
<=> 0 ≤ k ≤ 4 hay -0,05≤ k ≤1,94 hay -0,44 ≤ k ≤ 1,55
k = {0; 1; 2; 3; 4}
k = 0: x = 0 hay x = 1/2.arcsin(1/3) hay x = π/2 - 1/2.arcsin(1/3)
k = 1: x = π/2 hay x = 1/2.arcsin(1/3) + π hay x = 3π/2 - 1/2.arcsin(1/3)
k = 2: x = π
k = 3: x = 3π/2
k = 4: x = 2π = 0
Vậy tổng các nghiệm x = π/2 + π/2 + π + 3π/2 + π + 3π/2 = 6π