# How do you prove that csch^-1(x) = sinh^-1(1/x)?

##### 1 Answer

Jun 13, 2018

We seek to prove that:

# csch^(-1)(x) -= sinh^(-1)(1/x) #

Let:

# u = csch^(-1)(x) => cschu=x #

# v = sinh^(-1)(1/x) => sinhv=1/x #

Then we have:

# sinhv = 1/x = 1/(cschu) = sinhu #

And as

# sinhv = sinhu iff v = u #

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ iff sinh^(-1)(1/x) = csch^(-1)(x) \ \ \ # QED