We have to find the values of x in the interval [0, 2pi] that satisfy 4*cos(2x)*sin(2x) = 1
Use the formula for sin 2x which gives sin 2x = 2*sin x*cos x
Here we have 4*cos(2x)*sin(2x) = 1
=> 2*cos(2x)*sin(2x) = 1/2
=> sin 4x = 1/2
=> 4x = arc sin (1/2)
=> 4x = 30 degrees and 150 degrees
=> x = (1/4)*30 and (1/4)*150
=> x = 7.5 degrees and x = 37.5 degrees
The required solution is x = 7.5 degrees and x = 37.5 degrees
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}
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