# Math Help - Proof of inverse cosech(x)

1. ## Proof of inverse cosech(x)

How would you prove cosech^-1(x) = sinh^-1(-1/x)?

Both cosech and sinh are meant to mean inverse. Thanks in advance.

2. Originally Posted by JQ2009
How would you prove cosech^-1(x) = sinh^-1(-1/x)?

Both cosech and sinh are meant to mean inverse. Thanks in advance.
let $\sinh^{-1} \frac 1x = y$. solve for $x$ and find the $\text{csch}^{-1} x$ from that

Spoiler:

$\sinh^{-1} \frac 1x = y$

$\Rightarrow \frac 1x = \sinh y$

$\Rightarrow x = \frac 1{\sinh y} = \text{csch } y$

$\Rightarrow \text{csch}^{-1} x = y = \sinh^{-1} \frac 1x$