Distribution of $Z = \sin(X) \sin(Y)$ where $X$ and $Y$ are independent and uniform in $[-\pi,\pi]$?

Question

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 $Z$?

Answer 1

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)