Consider two random variables $X$ and $Y$ that are independent and uniformly distributed over a period, say $[-\pi,\pi]$. Which is the PDF (or the CDF if you prefer) of $Z = \sin(X) \sin(Y)$?
This is the "two-dimensional" case of the arcsine distribution, https://en.wikipedia.org/wiki/Arcsine_distribution , see also Distribution of sine of uniform random variable on $[0, 2\pi]$ . Is there some analytic method or this should be done numerically by sampling $Z$?
Hint:
Just find the distribution of difference of two arcsine distribution(which is much easier than a product)
$$Z= \sin(X)\sin(Y)=\frac{1}{2}\left(\cos(X-Y) -\cos(X+Y)\right) $$ $$\sim \frac{1}{2}\left(\sin(X-Y) -\sin(X+Y)\right) $$ $$\sim \frac{1}{2}\left(\sin(X) -\sin(Y)\right) $$
R code
n<-100000
x<-runif(n,-pi,pi)
y<-runif(n,-pi,pi)
plot(density(sin(x)*sin(y)))
lines(density(.5*(sin(x)-cos(y))),col=2)
If $U$ and $V$ are uniform so
all of the following random variables have same distribution
$\cos(U)$,$\cos(2U)$,$\sin(U)$,$\sin(2U)$,$\cos(U+V)$,$\cos(U-V)$,$-\cos(U)$
R code
n<-100000
x<-runif(n,-pi,pi)
y<-runif(n,-pi,pi)
plot(density(cos(x)))
lines(density(sin(y)),col=2)
lines(density(sin(2*x)),col=3)
lines(density(cos(2*x)),col=4)
lines(density(cos(x+y)),col=5)
lines(density(sin(x+y)),col=6)
