Simultaneous Hyperbolic Function Confusion

Question

I have a simultaneous hyperbolic problem I am completely perplexed on,

$\cosh(x)+\cosh(y)=4$

$\sinh(x)-\sinh(y)=2$

I manage to re write them all in terms of e;

$e^x+e^{-x}+e^y+e^{-y}=8$

$e^x-e^{-x}-e^y+e^{-y}=4$

However, I cannot think of a way to eliminate one of the variables.. Any hints?

Answer 1

Hint:

$$(4-\cosh y)^2-(2+\sinh y)^2=?$$

Use $\cosh^2y-\sinh^2y=1$

Now replace $\cosh y,\sinh y$ with $e^y,e^{-y}$ to form a quadratic equation in $e^y$

Note for real $y,e^y>0$