I am reading the book, Geometry of Special Relativity, by Tevian Gray.
In the introductory chapter to hyperbolic geometry, he states that the definition:
$$ \cosh(\beta) = \frac{e^\beta + e^{-\beta}}{2} $$
is equivalent to the definition: $$ \cosh(\beta) = x/\rho, $$ where $$ \rho = x^2-y^2 $$
These definitions make intuitive sense, but I'm not sure how to prove their equivalency. For context, I am an 18-year-old just out of high school, with a good grasp of simple variable calculus, but no exposure to college math yet.
The hyperbolic functions $\cosh$ and $\sinh$ are defined by $$ \cosh(\beta) = \frac{e^{\beta}+e^{-\beta}}{2}, \qquad \sinh(\beta) = \frac{e^{\beta}-e^{-\beta}}{2} $$ and satisfy the identity $\cosh(\beta)^2 - \sinh(\beta)^2 = 1.$
Now consider the hyperbola $x^2-y^2=\rho^2$ and rewrite it as $$ \left(\frac{x}{\rho}\right)^2 - \left(\frac{y}{\rho}\right)^2 = 1. $$ It has a clear similarity to the hyperbolic identity above and we can make the following identifications: $$ \frac{x}{\rho}=\cosh(\beta), \qquad \frac{y}{\rho}=\sinh(\beta). $$