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

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The discussion explores whether the expression Cosh(x) * Cosh(y) can be exclusively expressed in terms of k = Cosech(x) * Cosech(y). Despite attempts to manipulate hyperbolic identities, participants find that expressing Cosh(x) * Cosh(y) solely in terms of k leads to inconsistencies. Specifically, different values of sinh(a) and sinh(b) yielding the same k produce varying results for Cosh(x) * Cosh(y). This indicates that a single function f(k) cannot represent Cosh(x) * Cosh(y) uniquely for all cases. Ultimately, the conclusion is that it is not possible to rewrite Cosh(x) * Cosh(y) exclusively in terms of k.
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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.
 
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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).
 
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.
 
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.
 
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