The cartesian equation of the tractrix is:$$y=\pm\left(a\cdot \operatorname{arcsech}^{-1}\left(\frac{x}{a}\right)-\sqrt{a^2-x^2}\right)$$ where $a>0$ is a real parameter and $x$ varies from $0$ to $a$.
One parametric form is:$$\begin{cases}x(t)=a\cdot \operatorname{sech}(t)\\y(t)=a\cdot(t-\tanh(t)) \end{cases}$$
Why do the parametric form represent the same curve?; I mean why does the plus/minus sign disappear?
Because $y$ is no longer a function of $x$ in the parametric equations, we can have two points on the same vertical line (that is, different values of $y$ can occur for the same value of $x$). If $y$ is a function of $x$, we can only have one point on each vertical line, necessitating a $\pm$ sign. See vertical line test.