To solve the equation \(2 \cos x - \cos \frac{1}{2}x = 1\), we can use trigonometric identities and algebraic manipulation.

1. Use the double angle formula: \(\cos x = 2 \cos^2 \frac{x}{2} - 1\).

2. Substitute \(\cos x\) in the equation:

\(2(2 \cos^2 \frac{x}{2} - 1) - \cos \frac{x}{2} = 1\)

3. Simplify the equation:

\(4 \cos^2 \frac{x}{2} - 2 - \cos \frac{x}{2} = 1\)

4. Rearrange to form a quadratic equation in \(\cos \frac{x}{2}\):

\(4 \cos^2 \frac{x}{2} - \cos \frac{x}{2} - 3 = 0\)

5. Let \(u = \cos \frac{x}{2}\). The equation becomes:

\(4u^2 - u - 3 = 0\)

6. Solve the quadratic equation using the quadratic formula:

\(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

where \(a = 4\), \(b = -1\), \(c = -3\).

\(u = \frac{1 \pm \sqrt{1 + 48}}{8} = \frac{1 \pm 7}{8}\)

7. The solutions for \(u\) are:

\(u = 1\) and \(u = -\frac{3}{4}\).

8. Solve for \(x\):

For \(u = 1\), \(\cos \frac{x}{2} = 1\), so \(\frac{x}{2} = 0\) or \(x = 0\).

For \(u = -\frac{3}{4}\), \(\cos \frac{x}{2} = -\frac{3}{4}\), solve for \(\frac{x}{2}\):

\(\frac{x}{2} = \cos^{-1}(-\frac{3}{4})\), so \(x = 2 \cos^{-1}(-\frac{3}{4})\).

9. Calculate \(x\) for \(\cos^{-1}(-\frac{3}{4})\):

\(x \approx 4.84\).

Thus, the solutions are \(x = 0\) and \(x \approx 4.84\).