Solve for x on the interval [0, 2pi]: 4cos(2x)sin(2x)=1

4*cos(2x)*sin(2x) = 1

First
we will divide by 2 .

==> 2cos(2x)*sin(2x) =
1/2................(1)

Now we will use trigonometric
identities to solve.

We know that: sin2x =
2sinx*cosx

==> sin4x =
2sin(2x)*cos(2x)

Then we will substitute into
(1).

==> sin(4x) =
1/2

But we know that if sin(a)= 1/2 ==> a = pi/6 ,
and  5pi/6.

==> 4x = pi/6 ==> x =
pi/24

==> 4x = 5pi/6 ==> x =
5pi/24

Then the answer is:  x= { pi/24,
5pi/24}