I've been wondering for a while about how determinants in multiple variables could be expressed as a product of two newer and simplified determinants, in which case, the evaluation of individual determinants and the ultimate product is a lot easier than simplifying the original determinant.
For instance,
$\begin{vmatrix}cos(a-p) & cos(a-q) & cos (a-r)\\ cos(b-p) & cos(b-q) & cos(b-r)\\ cos(c-p) & cos(c-q) & cos(c-r)& \end{vmatrix}$
can be evaluated by expressing the elements with the trigonometric identity for cosine of (A-B), which however, is an unnecesarily long exercise.
However, the same determinant can be expressed as the product of two determinants as
$\begin{vmatrix}cos(a) & sin(a) & 0\\ cos(b) & sin(b) & 0\\ cos(c) & sin(c) & 0& \end{vmatrix}$.$\begin{vmatrix}cos(p) & sin(p) & 0\\ cos(q) & sin(q) & 0\\ cos(r) & sin(r) & 0& \end{vmatrix}$
which is evidently a lot easier to evaluate so, I was wondering is there was a way to use this kind of splitting method on other determinants. For instance,
$\begin{vmatrix}1 & ab & c(a+b)\\ 1 & bc & a(b+c)\\ 1 & ca & b(c+a)& \end{vmatrix}$.
Any kind of help/suggestion would be very appreciated.