I do know that $\operatorname{csch}(x) = \dfrac 1 { \sinh(x)}$, but I'm not sure if it applies to $x$ only. I don't know if it's applicable for $4x$ as well, or any other monomial.

# Is $1 / \sinh (4x)$ equal to $\operatorname{csch} (4x)$

hyperbolic-functionstrigonometry

## Best Answer

This is a good question that many people have a hard time grasping the first time around. I wouldn't worry about downvotes or close votes, as some of the users here sometimes misinterpret a question being 'simple' for a being a bad question.

In general, for any two functions $f(x)$ and $g(x)$ such that $f(x)=g(x)$ when $x$ is a member of some set/interval $x\in A$, given some transformation $$x\to u,$$ (Take $u=4x$, as in your case), if $u$ is also a member $u\in A$, then we should have that $$f(u)=g(u).$$

In your example, $A=\mathbb{R}\setminus\{0\}$, and for any $x\in\mathbb{R}\setminus\{0\}$, we also have $4x\in\mathbb{R}\setminus\{0\}$, thus this relationship is also true.