how many values of x insatisfy 2sin(1/2x)=cos(2x)?

We need to find the values of x in [0, 2*pi] that satisfy
2*(sin x/2) = cos 2x.

sin (x/2) = sqrt [(1 - cos
x)/2]

cos 2x = 2*(cos x)^2 -
1

As 2*(sin x/2) = cos
2x

=> 2*sqrt [(1 - cos x)/2] = 2*(cos x)^2 -
1

take the square of both the
sides

=> 4*(1 - cos x)/2 = 4*(cos x)^4 + 1 -
4*(cos)^2

=> 2 - 2*cos x = 4*(cos x)^4 + 1 -
4*(cos)^2

=> 4*(cos x)^4 - 4*(cos)^2 + 2*cos x - 1 =
0

We see that cos x has a highest power of 4. This gives us
4 values of cos x that we can get from solving the equation. Also, for each value of cos
x, there are two values of x that give the same value of cos
x.

This gives 8 values of x that can satisfy
the given equation.