cos4x=1-2sin^2(2x)=2-3sin2x,
2sin^2(2x)-3sin2x+1=0=(2sin2x-1)(sin2x-1).
So sin2x=1 or sin2x=1/2. Therefore 2x=π/2, and x=π/4, or 2x=π/6,and x=π/12.
In degrees this is x=45 or 15. Also, since sine is positive in Quadrant 2, 2x=π-π/6=5π/6 and x=5π/12=75 degrees.
CHECK
Substitute each of the 3 values of x into the original equation. We'll use degrees.
x=15: 2x=30; 4x=60; cos4x=1/2=2-3/2 OK
x=45: 2x=90; 4x=180; cos4x=-1=2-3 OK
x=75: 2x=150; 4x=300; cos4x=1/2=2-3/2 OK