1. Start with the equation:

\(\cot \theta - \cot(\theta + 45^\circ) = 3\)

2. Use the identity \(\cot(\theta + 45^\circ) = \frac{1 - \tan \theta}{\tan \theta + 1}\).

3. Substitute \(\cot \theta = \frac{1}{\tan \theta}\) and \(\cot(\theta + 45^\circ) = \frac{1 - \tan \theta}{\tan \theta + 1}\) into the equation:

\(\frac{1}{\tan \theta} - \frac{1 - \tan \theta}{\tan \theta + 1} = 3\)

4. Simplify and rearrange to form a quadratic equation:

\(1 + \tan \theta - \tan \theta(1 - \tan \theta) = 3 \tan \theta(1 + \tan \theta)\)

5. Expand and simplify:

\(2\tan^2 \theta + 3\tan \theta - 1 = 0\)

6. Solve the quadratic equation using the quadratic formula:

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

where \(a = 2\), \(b = 3\), \(c = -1\).

7. Calculate:

\(\tan \theta = \frac{-3 \pm \sqrt{3^2 - 4 \times 2 \times (-1)}}{2 \times 2}\)

\(\tan \theta = \frac{-3 \pm \sqrt{9 + 8}}{4}\)

\(\tan \theta = \frac{-3 \pm \sqrt{17}}{4}\)

8. Find \(\theta\) for \(0^\circ < \theta < 180^\circ\):

\(\theta = 15.7^\circ\) or \(\theta = 119.3^\circ\).