problem regards hyperbolic function

Hi I have been having a problem regards this question. I spent some time trying to sit through it but to no result.

The Question is

Show that

arccosech x = log((1+squareroot(1+x^2)) for x>0

log((1-squareroot(1+x^2)) for x<0

Justify the sign of any square roots.

So this is what I did

I assume that y=arccosech(x) thereby cosech(y)=x

1/(sinh(y)) = 1/(1/2*(e^y)-(e^-y))

2/x=e^y - e^-y

In(2/x)=In(e^y)-In(e^-y)

In(2/x)=2y

2arccosechx=In(2/x)

this is totally wrong but I do not know how to continue or to handle problem such as this one. Can any one give me a hint and direction about solving this problem?

Best Regards

Junks

Re: problem regards hyperbolic function

Re: problem regards hyperbolic function

thank you for your reply Felix, yes I did made a mistake. I forgot to put -In X into the equation. Just one small question though, so I subbed in U=e^y as you suggested, I did some algebra and got the equation to become u^2 - 2u/x -1 =0 I use quadratic formula for u, (-b plus of minus squareroot of (b^2 -4ac))/2a. The problem is I got U = (2/x +1) and U=-1, which is wrong. Do you know what I did wrong?

Re: problem regards hyperbolic function