I haven’t done geometry in a while, but I graphed both in Apple’s handy graphing calculator, and it appears that they don’t intersect at any points other than the vertex, assuming I graphed 'em right. But, for the “why”, you’ll have to ask someone with a better memory than I…

enolancooper, I’m sure you know what the form of cosh(x) is, so I’m not trying to inform you, but I do want to clear one thing up for anyone who doesn’t.

cosh(x) = (e[sup]x[/sup] + e[sup]-x[/sup])/2

What you have listed there is this:

(e[sup]2[/sup] - e[sup]-2[/sup])/2 = sinh(2)

And while the others here has answered the question very well, I would like to add that if you want to find the points of intersection without a graphing calculator, you can do so by finding the zero of this function:

cosh(x) - 1 - x[sup]2[/sup]

Newton’s Method is the simplest way I know. Start with some x, say 3, and keep taking the value

x - (cosh(x) - 1 - x[sup]2[/sup])/(sinh(x) - 2x)

and using that for your new value of x. The numerator is your function and the denominator is the derivative of your function. Soon you’ll get the answer. I did it, and here’s what I got: