parametric equation for hyperbola like $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Question

How can I get this parametric equation $x=\frac a2\left(t+\frac1t\right)$ and $y=\frac b2\left(t-\frac1t\right)$ for hyperbola like $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ using this $xy=1$ ($x=t$, $y=\frac 1t$, $t\neq 0$)

Answer 1

The parametric point for a hyperbola can be given by (asecθ, btanθ) where you can put cot(θ/2) as t and get your result