Finding probability distribution of quantity depending on other distributions

I have a vector that depends on the coordinates of randomly drawn unit vectors in ${\mathbb{R}}^{2}$:

$\sigma =\left(\begin{array}{c}\frac{\mathrm{cos}({\theta}_{a})+\mathrm{cos}({\theta}_{b})}{1+\mathrm{cos}({\theta}_{a})\mathrm{cos}({\theta}_{b})}\\ \frac{\mathrm{sin}({\theta}_{a})+\mathrm{sin}({\theta}_{b})}{1+\mathrm{sin}({\theta}_{a})\mathrm{sin}({\theta}_{b})}\end{array}\right)$

Here, ${\theta}_{a},{\theta}_{b}\in [0,2\pi )$, are drawn uniformly random from the unit circle, and are thus angles that parametrize unit vectors in ${\mathbb{R}}^{2}$. I am interested in figuring out a probability distribution for the coordinates of the $\sigma $-vector, but I am not well-versed in probability theory. I have been told that it is possible to somehow find a probability distribution for a quantity that depends on other distributions by somehow using the Jacobian matrix, but I have been unable to figure out how on my own.

My questions are thus:

Is there a method for finding a probability distribution for a quantity that depends on other distributions? If so, how? Does it have a name so I can learn about form other sources?

If the method exists, how would I use it on my concrete example?

I have a vector that depends on the coordinates of randomly drawn unit vectors in ${\mathbb{R}}^{2}$:

$\sigma =\left(\begin{array}{c}\frac{\mathrm{cos}({\theta}_{a})+\mathrm{cos}({\theta}_{b})}{1+\mathrm{cos}({\theta}_{a})\mathrm{cos}({\theta}_{b})}\\ \frac{\mathrm{sin}({\theta}_{a})+\mathrm{sin}({\theta}_{b})}{1+\mathrm{sin}({\theta}_{a})\mathrm{sin}({\theta}_{b})}\end{array}\right)$

Here, ${\theta}_{a},{\theta}_{b}\in [0,2\pi )$, are drawn uniformly random from the unit circle, and are thus angles that parametrize unit vectors in ${\mathbb{R}}^{2}$. I am interested in figuring out a probability distribution for the coordinates of the $\sigma $-vector, but I am not well-versed in probability theory. I have been told that it is possible to somehow find a probability distribution for a quantity that depends on other distributions by somehow using the Jacobian matrix, but I have been unable to figure out how on my own.

My questions are thus:

Is there a method for finding a probability distribution for a quantity that depends on other distributions? If so, how? Does it have a name so I can learn about form other sources?

If the method exists, how would I use it on my concrete example?