Given:
\begin{equation*}
\frac{a}{b}=\frac{c}{d}
\end{equation*}
Componendo:
\begin{equation*}
\frac{a+b}{b}=\frac{c+d}{d}
\end{equation*}
Dividendo:
\begin{equation*}
\frac{a-b}{b}=\frac{c-d}{d}
\end{equation*}
The proofs are “obvious”. You are just adding/subtracting 1 to both sides.
What’s more interesting is combining the two. Divide the first by the second to get:
\begin{equation*}
\frac{a+b}{a-b}=\frac{c+d}{c-d}
\end{equation*}