Find all variables for 2-3sin2x=cos4x within the interval [0,2(3.14)

Question

Cant seem to find the answer for this question or the correct identity to even begin with so am seeking help... please.

Answer 1

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