1. Use the identity \(\cot 2x = \frac{\cot^2 x - 1}{2\cot x}\) to express \(\cot 2x\) in terms of \(\cot x\).

2. Substitute into the equation: \(\frac{\cot^2 x - 1}{2\cot x} + \cot x = 3\).

3. Multiply through by \(2\cot x\) to clear the fraction: \(\cot^2 x - 1 + 2\cot^2 x = 6\cot x\).

4. Simplify to get: \(3\cot^2 x - 6\cot x - 1 = 0\).

5. Let \(y = \cot x\), then the equation becomes \(3y^2 - 6y - 1 = 0\).

6. Solve the quadratic equation using the quadratic formula: \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3, b = -6, c = -1\).

7. Calculate the discriminant: \(b^2 - 4ac = 36 + 12 = 48\).

8. Solve for \(y\): \(y = \frac{6 \pm \sqrt{48}}{6}\).

9. Simplify to find \(y = 3 \pm \sqrt{3}\).

10. Find \(x\) using \(x = \cot^{-1}(y)\).

11. Calculate \(x\) for \(y = 3 + \sqrt{3}\) and \(y = 3 - \sqrt{3}\) within the interval \(0^\circ < x < 180^\circ\).

12. The solutions are \(x = 24.9^\circ\) and \(x = 98.8^\circ\).