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

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.

Solution 1:

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.

Solution 2:

Graviton's answer is entirely correct, but may be difficult to understand for someone with less mathematical experience, so I am going to offer a simpler way of phrasing the same thing.

The problem here is that you actually have two different variables which are both called $x$. That's confusing, and in mathematics, we try to avoid this confusion, usually by renaming one of the variables to fix the clash. In this case, we want to rename the variable belonging to the identity, because the identity is supposed to be a universal rule (whereas the $4x$ that appears in your problem belongs to some specific context, and so if we go renaming that, we would need to rewrite the whole problem to match).

So, let's start over. We know that $\mathrm{csch}(u) = \frac{1}{\mathrm{sinh}(u)}$, and we have some problem which involves one of $\mathrm{csch}(4x)$ or $\frac{1}{\mathrm{sinh}(4x)}$. Can we substitute the one for the other? Are we allowed to write "Let $u = 4x$" and move on with it?

In general, yes, because it's supposed to be a universal rule that works for all values of $u$. But there are a couple of caveats to keep in mind:

• We cannot select a $u$ which will result in $\mathrm{sinh}(u) = 0$, because then we would be dividing by zero. This means we cannot have $x = 0$ if we want $u = 4x$. If zero is a possible value of $x$, then you will need to do something about that, because zero doesn't have a hyperbolic cosecant. There are also some complex numbers at which this happens, but you might not care if you're only working in the reals.
• More generally, we must select a $u$ at which both sides of the identity are well-defined, and where the identity is known to be valid. The "don't divide by zero" rule is just a special case of this. But you can also run into trouble with a variety of other identities, such as those involving square roots and logarithms. Here's a nasty example of that problem:

$$ -1 = \sqrt{-1}\sqrt{-1}\color{red}{\neq}\sqrt{-1 \times -1} = \sqrt{1} = 1 $$

The identity $\sqrt{a}\sqrt{b} = \sqrt{ab}$ is entirely valid - within the real numbers. But within the real numbers, there's no such thing as a square root of a negative number, and it turns out that this identity does not fully generalize to the complex numbers (specifically, if you think of multiplication as a form of scaling and rotation, the identity only works when the product does not rotate through the negative real axis in either direction). So you need to keep track of whether you are staying within the real numbers (where the identity always works, but you can't take the square roots of negative numbers in the first place), or working in the complex numbers (where you can take the square root of whatever you want, but the identity has strings attached).

To summarize: Yes, you can use this particular identity for $4x$ (if given $x \neq 0$), for any monomial, and indeed for any real or complex expression under the sun, as long as it's nonzero (for complex expressions: as long as it isn't an integer multiple of $\pi i$). But some identities have stricter rules than that, and you do need to verify that a given identity actually applies to a given scenario before using it.