I know of the identity $\displaystyle \cosh^2 t - \sinh^2 t = 1$, which directly leads to hyperbola equation, but this isn't what is expected here.The term "hyperbolic" coems from the fact that if we define the Cartesian coordinates of a point to be $\displaystyle x=\cosh t$ and $\displaystyle y=\sinh t$, where $\displaystyle t$ is so-calledparametric variable, then elimination of $\displaystyle t$ leads to the equation of $\displaystyle x^2-y^2=1$, the equation of an hyperbola.

So, how do you "eliminate $\displaystyle t$"?