Hyperbolic Inverses?

[Assassin315]

Ugh, I just don't get these hyperbolic ones.

This question says "Use the fact that tanh(y) = x to express the inverse function y = arctanh(x) in the form:

y = arctanh(x) = .5ln((1+x)/(1-x))

I am absolutely clueless how to start proving this identity. Maybe it's because I don't know some hyperbolic identities, but I can't get anywhere with this. Can I get a nudge in the right direction?

 

[galactus]

Let \(\displaystyle y=tanh^{-1}(x)\)

Then, \(\displaystyle x=tanh(y)=\frac{e^{y}-e^{-y}}{e^{y}+e^{-y}}=\frac{e^{2y}-1}{e^{2y}+1}\)

\(\displaystyle xe^{2y}+x=e^{2y}-1\)

There's the nudge. Now continue, solve for y and you have it.