The following is a derivation of a density from a paper I am currently studying. Sorry for the bad quality, it is quite an old paper. I need to clarify that $R$ has the standard exponential density in $(0,\infty)$, $U$ is uniform on $(0,1)$ and they are independent. The population correlation coefficient $\rho$ is a constant of course. $X$ and $Y$ come from the standard bivariate normal distribution, hence the trigonometric representation, but this plays no role here, I believe.
What I do not understand is how the author reaches these conclusions for positive or negative $t$. It seems to me that the division by a negative number and the nonnegativity of $R$ are not properly taken into account. I could be mistaken of course so I would appreciate some advice. Thank you.