One of the standard equations for a rectangular "reciprocal" hyperbola which is in the $1$st and $3$rd quadrants is $$xy=c^2$$
And its corresponding parametric equations are $x = ct$ and $y = \frac{c}{t}$, where $t$ can be any real number, except $0$.
Another standard form of the rectangular hyperbola is "horizontal" or "left-right", i.e. where the its transverse axis is $\parallel$ to the $x$-axis. Its equations can be derived by rotating the "reciprocal" hyperbola ($xy=c^2$) clockwise by $\frac{\pi}{4}$.
\begin{align} \begin{pmatrix} \cos\frac{\pi}{4} & \sin\frac{\pi}{4} \\ -\sin\frac{\pi}{4} & \cos\frac{\pi}{4} \end{pmatrix} \begin{pmatrix} ct \\ \frac{c}{t} \end{pmatrix} = \begin{pmatrix} \frac{ct\sqrt{2}}{2} + \frac{c\sqrt{2}}{2t} \\ \frac{c\sqrt{2}}{2t} - \frac{ct\sqrt{2}}{2} \end{pmatrix} \end{align}
So the parametric equations are
\begin{align} \begin{cases} x = \frac{ct\sqrt{2}}{2} + \frac{c\sqrt{2}}{2t} \\ y = \frac{c\sqrt{2}}{2t} - \frac{ct\sqrt{2}}{2} \end{cases} \end{align}
However, the parametric equations can also be derived geometrically in an alternative form:
\begin{align} \begin{cases} x = a \sec t \\ y = a \tan t \end{cases} \end{align}
Where (I believe) $a = c\sqrt{2}$. How can we show the equivalence of these forms or use one to derive another algebraically?