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.
I agree that your method works, however in my opinion it has two disadvantages:
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.