Solving hyperbolic trigonometric equations

[scotty_le_b]

Homework Statement

Show that the real solution $x$ of $$tanhx=cosechx$$ can be written in the form $x=ln(a \pm \sqrt{a})$ and find an explicit value for $a$.

Homework Equations

$$cosh^{2}x-sinh^{2}x=1$$
$$coshx=\frac{e^{x}+e^{-x}}{2}$$

The Attempt at a Solution

I reduced the original equation $$tanhx=cosechx$$ to $$0=cosh^{2}x-coshx-1$$ I used the quadratic formula to get $coshx= \frac{1 \pm \sqrt{5}}{2}$. I discarded $\frac{1- \sqrt{5}}{2}$ since $coshx$ must be greater than 0. I then expanded the solution to $$e^{x}+e^{-x}=1+ \sqrt{5}$$ I rearranged this to a quadratic in $e^{x}$ and solved using the quadratic formula and then took logs of both sides to get: $$x=ln\Bigg(\frac{1+ \sqrt{5}}{2} \pm \sqrt{\frac{1+ \sqrt{5}}{2}}\Bigg)$$ Obviously from this the solution can be expressed in the form $x=ln(a \pm \sqrt{a})$ when $a= \frac{1+ \sqrt{5}}{2}$. I feel like this is cheating though since the question asks me to show the solution can be expressed in a specific form and then find the solution. Is it acceptable to find the solution and show it can be expressed in that form even with the wording of the question? I couldn't think of any other way to show the solution could be expressed in that form without explicitly solving it first.[/B]

 

[PeroK]

It's hardly cheating. I suspect it's just the wording of the question.