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