\( A = \{ x : - \frac { 2 \pi } { 3 } < x < \frac { 2 \pi } { 3 } \wedge \cos ( 2 x - \frac { \pi } { 3 } ) = \sin ( x ) \} \)

To solve the equation \( \cos \left( 2x - \frac{\pi}{3} \right) = \sin(x) \) within the bounds \( -\frac{2\pi}{3} < x < \frac{2\pi}{3} \), we can first apply a trigonometric identity. We know that \( \sin(x) = \cos\left(\frac{\pi}{2} - x\right) \). This suggests we can rewrite the given equation in terms of cosines. Next, let's explore the angle difference identity for cosine: \[ \cos(2x - \frac{\pi}{3}) = \cos(2x) \cos(\frac{\pi}{3}) + \sin(2x) \sin(\frac{\pi}{3}). \] This simplifies to: \[ \frac{1}{2} \cos(2x) + \frac{\sqrt{3}}{2} \sin(2x) = \sin(x). \] Now, substituting \( \sin(x) \) using the same approach, we will have a manageable equation to solve for \( x \). Now, backtracking to the original constraints, you can also examine specific values of \( x \) in the specified interval, such as testing \( x = 0 \) or \( x = \frac{\pi}{6} \), to find potential solutions that satisfy both the equation and the interval restrictions. Using a graphical approach by plotting the functions can also guide towards where these curves intersect within the interval bounds.