Question

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

Answer 1

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 `sinh` is a `1:1` function we conclude that:

`sinhv = sinhu iff v = u`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ iff sinh^(-1)(1/x) = csch^(-1)(x) \ \ \` QED