Is there any use for $\sin(\sin x)$?

The intensity of light diffracted at a slit as a function of the angle actually involves a term $\sin\left(\frac{\alpha\beta}{2}\sin(\theta)\right)$, see

https://en.wikipedia.org/wiki/Fraunhofer_diffraction

(I'm no physicist at all, but this has been stuck in my head since high school just because it is such an unusual term to encounter naturally)

People in complex dynamics consider the behavior of all sorts of functions under iteration. For example, here is the Julia set of $\sin(z)$.

In that context, it makes perfect sense to talk about $\sin(\sin(\cdots \sin(z)))$.

Since $\sinh(x) = i\sin(i x)$ is the odd part of the exponential function, we can interpret it (for example within the framework of combinatorial species) as the (exponential) generating function for sets of odd size.

Thus, $\sinh(\sinh(x)) = -i\sin(\sin(ix))$ is the (exponential) generating function for set partitions with an odd number of parts, each of which has odd cardinality.

We can slightly refine this by interpreting $\sin(\omega\sin(x))$ as the generating function of a weighted species, giving each set partition the weight $(-1)^{n-1} w^b$, where $b$ is the number of blocks and $n$ is the size of the ground set.

Similarly, $\cosh(\sinh(x))$ is the generating function for set partitions with an even number of parts, all of which are of odd cardinality, and $\cosh(\cosh(x)-1)$ is the generating function for set partitions with an even number of (nonempty) parts, all of which are of even cardinality. Note however, that the coefficients of $\cos(\cos(x)-1)$ and $\cosh(\cosh(x)-1)$ are very different, whereas the coefficients of $\sin(\sin(x))$ and $\sinh(\sinh(x))$ only differ in sign.

As an aside, $\sin(\sin(\cdot))$ satisfies a nice differential equation: $$ (f^2-1)^2 (f''' + f') - 3 (f^2-1) f f' f'' + (2f^2+1) f'^3 = 0 $$ while $\sinh(\sinh(\cdot))$ satisfies: $$ (f^2+1)^2 (f''' - f') - 3 (f^2+1) f f' f'' + (2f^2-1) f'^3 = 0 $$