trig question pls help

Question

find the solution set of sin²(4x)+5cos²(2X)-2=0 given the interval [0,2pi]

Use only trigonometric identities.

Answer 1

sin²(4x) + 5cos²(2X) - 2=0; 1 - cos²(4x) + 5(1 + cos(4x))/2 - 2 = 0; 2 - 2cos²(4x) + 5 + 5cos(4x) - 4 = 0;

2cos²(4x) - 5cos(4x) - 3 = 0; (cos(4x) - 3)(2cos(4x) + 1) = 0; cos(4x) = - 3, no solutions;

cos(4x) = -1/2; 4x = 2π/3 + 2πn or 4x = 4ππ/3 + 2πn;

x = π/6 + πn/2 or x = π/3 + πn/2, n⊂Z