Prove that the inverse hyperbolic functions are the following logarithms: a. (cosh ^{-1} x=ln left(x+sqrt{x^{2}-1} ight)). b. (tanh ^{-1} x=frac{1}{2} ln frac{1+x}{1-x}).

Prove that the inverse hyperbolic functions are the following logarithms:

a. \(\cosh ^{-1} x=\ln \left(x+\sqrt{x^{2}-1}\right)\).

b. \(\tanh ^{-1} x=\frac{1}{2} \ln \frac{1+x}{1-x}\).