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

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

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

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

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