Can cosh(x)cosh(y) be rewritten in terms of k=Cosech(x)*Cosech(y)?

[emanuele.levi]

Given the quantity

Cosh(x)*Cosh(y)


where x and y are two indipendent real variables is it possible to write it only in function of


k=Cosech(x)*Cosech(y)

?
It could seem a quite easy problem but I spent a few days between the proprieties of hyperbolic functions and I really didn't find a way to solve it.

 

[Russell Berty]

Hint:
Use the identity (cosh(t))^2-(sinh(t))^2=1 to solve for cosh(t).

Then use the fact that csch(t) = 1/sinh(t) so sinh(t) = 1/csch(t).

 

[emanuele.levi]

thank you for the hint Russell,
but that's not a solution to my problem, as I want to write the quantity

Cosh(x)*Cosh(y)


ONLY in function of k. If I did like you suggested me, I find terms like


Sinh(x)+Sinh(y)


and I can't find a way to write them in function of k.

 

[Russell Berty]

It is not possible.
Assume you have some function f(k) that represents cosh(x)cosh(y) in terms of k.
When k = 1/2, then what would f(k) be?

Let sinh(a)=.5, sinh(b)=4. Then k = 1/(.5*4) = 1/2
Then cosh(a)*cosh(b)=sqr( 1+1/4)*sqr( 1+16)=sqr(85/4)=f(1/2)


However look at:
let sinh(c)=1, sinh(d)=2. Then k = 1/(1*2) = 1/2
But cosh(a)*cosh(b)=sqr( 1+1)*sqr( 1+4)=sqr(10)=f(1/2)

So, f(1/2) would not have a single output value, it is not a function.