Suppose that \(Y_{1}, \ldots, Y_{n}\) are independent and identically distributed with density

\[
\frac{1}{2 \pi}(1+\psi \cos y)(1+\lambda \sin y)
\]

on the interval \(-\pi
\[
L(\psi, \lambda ; y)=L_{1}(\psi ; y) \times L_{2}(\lambda ; y) .
\]

Show that the likelihood factorization is also a density factorization, i.e., for fixed \(\psi\), that \(L_{1}(\psi ; y)\) is a probability density on \((-\pi, \pi)^{n}\), and likewise for \(L_{2}\). Does it follow that \(R\) and \(S\) are independent?