\((\csc\theta + \cot\theta)(\csc\theta - \cot\theta) = \csc^2\theta - \cot^2\theta\)
\(= (1 + \cot^2\theta) - \cot^2\theta\)
\(= 1\)

\(\Rightarrow \csc\theta + \cot\theta = \dfrac{1}{\csc\theta - \cot\theta}\)

\(\dfrac{\tan^2\theta + (1+\sec\theta)^2}{\tan\theta(1+\sec\theta)}\)
\(= \dfrac{\tan^2\theta + 1 + 2\sec\theta + \sec^2\theta}{\tan\theta(1+\sec\theta)}\)
\(= \dfrac{\sec^2\theta + 2\sec\theta + \sec^2\theta}{\tan\theta(1+\sec\theta)}\)
\(= \dfrac{2\sec\theta(1+\sec\theta)}{\tan\theta(1+\sec\theta)}\)
\(= \dfrac{2\sec\theta}{\tan\theta}\)
\(= 2 \cdot \dfrac{\frac{1}{\cos\theta}}{\frac{\sin\theta}{\cos\theta}}\)
\(= \dfrac{2}{\sin\theta}\)

\(\sec\theta - \tan\theta = \dfrac{1}{\cos\theta} - \dfrac{\sin\theta}{\cos\theta}\)
\(= \dfrac{1 - \sin\theta}{\cos\theta}\)

\((\sec\theta - \tan\theta)^2 = \dfrac{(1 - \sin\theta)^2}{\cos^2\theta}\)
\(= \dfrac{(1 - \sin\theta)^2}{1 - \sin^2\theta}\) \(\;\) [since \(\cos^2\theta = 1 - \sin^2\theta\)]
\(= \dfrac{(1 - \sin\theta)^2}{(1 - \sin\theta)(1 + \sin\theta)}\)
\(= \dfrac{1 - \sin\theta}{1 + \sin\theta}\)

\(\dfrac{\cot\theta}{1+\cot^2\theta} = \dfrac{\cot\theta}{\csc^2\theta}\) \(\;\) [since \(1+\cot^2\theta = \csc^2\theta\)]
\(= \cot\theta \cdot \sin^2\theta\) \(\;\) [since \(\dfrac{1}{\csc^2\theta} = \sin^2\theta\)]
\(= \dfrac{\cos\theta}{\sin\theta} \cdot \sin^2\theta\)
\(= \cos\theta \sin\theta\)

\(\dfrac{1}{\cos^2\theta} + \dfrac{1}{\sin^2\theta} = \dfrac{\sin^2\theta + \cos^2\theta}{\sin^2\theta \cos^2\theta}\)
\(= \dfrac{1}{\sin^2\theta \cos^2\theta}\) \(\;\) [since \(\sin^2\theta + \cos^2\theta = 1\)]
\(= \dfrac{1}{\cos^2\theta} \cdot \dfrac{1}{\sin^2\theta}\)
\(= \sec^2\theta \csc^2\theta\)

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

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

\( \cos\theta \approx 0.4342 \) 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 \( 315^\circ \)

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