How do you derive the identity of inverse tanh using it's definition?

Question

The identity I'm referring to was 1/2*ln((1+x)/(1-x)), |x|<1

Answer 1

y= (e^x-e^-x)/(e^x+e^-x)
call e^x=z
y=( z^2-1)/(z^2+1)
so y*z^2+y= z^2-1
(y-1)z^2=-1-y and z = sqrt (1+y)/(1-y) z=e^x is `positive
so x= 1/2 ln (1+y)/(1-y)
interchange x and y
and
y= 1/2 ln (( 1+x)/(1-x))