(a) Start with the equation:

\(\cot 2\theta + \cot \theta = 2\)

Using the identity \(\cot 2\theta = \frac{1 - \tan^2 \theta}{2\tan \theta}\), substitute into the equation:

\(\frac{1 - \tan^2 \theta}{2\tan \theta} + \frac{1}{\tan \theta} = 2\)

Combine the terms over a common denominator:

\(\frac{1 - \tan^2 \theta + 2}{2\tan \theta} = 2\)

Simplify:

\(\frac{3 - \tan^2 \theta}{2\tan \theta} = 2\)

Multiply through by \(2\tan \theta\):

\(3 - \tan^2 \theta = 4\tan \theta\)

Rearrange into a quadratic equation:

\(\tan^2 \theta + 4\tan \theta - 3 = 0\)

(b) Solve the quadratic equation \(\tan^2 \theta + 4\tan \theta - 3 = 0\) using the quadratic formula \(\tan \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1, b = 4, c = -3\):

\(\tan \theta = \frac{-4 \pm \sqrt{16 + 12}}{2}\)

\(\tan \theta = \frac{-4 \pm \sqrt{28}}{2}\)

\(\tan \theta = \frac{-4 \pm 2\sqrt{7}}{2}\)

\(\tan \theta = -2 \pm \sqrt{7}\)

Calculate \(\theta\) for \(\tan \theta = -2 + \sqrt{7}\):

\(\theta = \tan^{-1}(-2 + \sqrt{7}) \approx 0.573\)

Calculate \(\theta\) for \(\tan \theta = -2 - \sqrt{7}\):

\(\theta = \tan^{-1}(-2 - \sqrt{7}) \approx 1.783\)

Thus, the solutions are \(\theta \approx 0.573\) and \(\theta \approx 1.783\).