\(\dfrac{\sec x}{\csc^2 x} - \dfrac{\csc x}{\sec^2 x} = (1 + \cot x + \tan x)(\sin x - \cos x)\)
LHS
\begin{equation*}
\begin{split}
& \frac{\sec x}{\csc^2 x} - \frac{\csc x}{\sec^2 x} \\\\
& \frac{\sec^3 x - \csc^3 x}{\sec^2 x \csc^2 x} \quad {\color {blue} \frac{\sin^3 x \cos^3 x}{\sin^3 x \cos^3 x}}\\\\
& \frac{\sin^3 x - \cos^3 x}{\sin x \cos x} \quad {\color {blue} a^3 - b^3 = (a - b)(a^2 + ab + b^2)} \\\\
& \frac{(\sin x - \cos x)(\sin^2 x + \sin x \cos x + \cos^2 x)}{\sin x \cos x)} \\\\
& (\sin x - \cos x)(\tan x + 1 + \cot x) \ce{->} \textbf{RHS}
\end{split}
\end{equation*}
LHS = RHS