Find the value of \( x \) if \( \cos(2x) = \frac{1}{2} \) and \( x^3 + 1 = 0 \)

Step 1 ：First we solve the equation \( \cos(2x) = \frac{1}{2} \). The general solution to this equation is \( x = \frac{\pi}{3} + 2n\pi \) or \( x = -\frac{\pi}{3} + 2n\pi \), where \( n \) is an integer.

Step 2 ：Next, we solve the equation \( x^3 + 1 = 0 \). This is a sum of cubes equation. It can be factored into \( (x+1)(x^2 - x + 1) = 0 \). Setting each factor equal to zero gives us \( x = -1 \) or \( x^2 - x + 1 = 0 \). The second equation has no real solutions, so \( x = -1 \) is the only solution to this equation.

Step 3 ：The final step is to find the common solutions to both equations. The only common solution is \( x = -1 \), which corresponds to \( n = -1 \) in the first equation.