\( \dfrac{1}{\cos\theta} = 3\cos\theta + 1 \)
\( 1 = 3\cos^2\theta + \cos\theta \)
\( 3\cos^2\theta + \cos\theta - 1 = 0 \)

\( \cos\theta = \dfrac{-1 \pm \sqrt{1^2 - 4(3)(-1)}}{2 \times 3} \)
\( = \dfrac{-1 \pm \sqrt{13}}{6} \)

\( \cos\theta \approx 0.4342 \) or \( \cos\theta \approx -0.7675 \)

For \( \cos\theta = 0.4342 \): \( \theta = 64.3^\circ \) and \( 360^\circ - 64.3^\circ = 295.7^\circ \)
For \( \cos\theta = -0.7675 \): \( \theta = 140.1^\circ \) and \( 360^\circ - 140.1^\circ = 219.9^\circ \)

\(\boxed{\theta = 64.3^\circ,\; 140.1^\circ,\; 219.9^\circ,\; 295.7^\circ}\)

\( 4\cot^2\theta - 2\cot\theta = 3(1 + \cot^2\theta) \)
\( 4\cot^2\theta - 2\cot\theta - 3 - 3\cot^2\theta = 0 \)
\( \cot^2\theta - 2\cot\theta - 3 = 0 \)

\( (\cot\theta - 3)(\cot\theta + 1) = 0 \)
\( \cot\theta = 3 \) or \( \cot\theta = -1 \)

\( \tan\theta = \dfrac{1}{3} \) or \( \tan\theta = -1 \)

For \( \tan\theta = \dfrac{1}{3} \): \( \theta = 18.43^\circ \) and \( 180^\circ + 18.43^\circ = 198.43^\circ \)
For \( \tan\theta = -1 \): \( \theta = 135^\circ \) and \( 315^\circ \)

\(\boxed{\theta = 18.4^\circ,\; 135^\circ,\; 198.4^\circ,\; 315^\circ}\)