I know of the identity , 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 and , where is so-called parametric variable, then elimination of leads to the equation of , the equation of an hyperbola.
So, how do you "eliminate "?
I believe that you are supposed to use the defining equations for cosh(t) and sinh(t), the ones in terms of the expontials e^t and e^(-t).
Start with (cosh t)^2-(sinh t)^2, and substitute the exponential definitions for these expression. Multiply out, simplify, and then you will get 1. This says that these functions satisfy the equation for the hyperbola.
1. The approach is more difficult than it needs to be, and
2. You need to know the answer in advance. Extremely unlikely in an exam situation.
Posts #2 and #4 say exactly how to do it - quick and simple.