unit hyperbola

The unit hyperbola (cf. the unit circle) is the special case

$$x^2-y^2=1$$

of the hyperbola

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

where both the $a$ and the $b$ have equal to 1.  The unit hyperbola is rectangular, i.e. its asymptotes ($y=\pm x$) are at right angles to each other.

The unit hyperbola has the well-known parametric

$$x=\pm \cosh t,y=\sinh t,$$

and also a trigonometric

$$x=\sec t,y=\tan t.$$

The former yields the rational

$$x=\frac{u^2+1}{2u},y=\frac{u^2-1}{2u}$$

when one substitutes  $e^t=u$, and the latter, via the substitution  $\tan \frac{t}{2}=u$, the rational

$$x=\frac{1+u^2}{1-u^2},y=\frac{2u}{1-u^2}$$

(which does not give the left apex of the hyperbola).