The equation is 2 cos(2x) - √3 = 0 and interval is [ 0, 2π ).

cos(2x) = √3/2

Let 2x = t, the function cos(t) has a period of 2π, first find all solutions in the interval [ 0, 2π ).

cos(t) = √3/2

cos(t) = cos(π/6).

The genaral solution of cos(x) = cos(α) is x = 2nπ ± α, where n is an integer. 

t = 2nπ ± π/6

2x = 2nπ ± π/6

x = nπ ± π/12

x = π(n ± 1/12)

x = π(n - 1/12) or x = π(n + 1/12).

The solutions outside the interval [ 0, 2π ) are

If n = 0 then x = π[ (0) - 1/12 ] = - π/12 < 0

If n = 2 then x = π[ (2) + 1/12 ] = 25π/12 > 0.

The solutions in the interval [ 0, 2π ) are

x = π[ (0) + 1/12 ] = π/12,

x = π[ (1) - 1/12 ] = 11π/12,

x = π[ (1) + 1/12 ] = 13π/12,

x = π[ (2) - 1/12 ] = 23π/12.