Why do I have to learn all these tricks for 'factoring' 3x3 matrices determinants?

$[br][br]sinx I cosx I 1[br]sin^2x | cos^2x |1[br]sin^3x | cos^3x |1[br]$
no modulus just trying to make the matrices lines clearer

And what exactly seems to be the problem?

Have you found the determinant of that matrix?

just expanded it and factored it into
$[br]sinxcosx(1-sinx)(1-cosx)(cosx-sinx)[br]$

if I tried to carefully notice and think throguh all these tricks to factor it it would have taken me like 45 minutes rofl, expanding it only takes like 10

Expanding down the third column would be much easier. By doing this, you are only multiplying each minor (the 2x2 matrices) by 1.

yeah I did that. I just expanded the matrice and factored it which was what i thought would be the logical way to do it, but apparently I wasn't supposed to do it like that

I would just leave it without factorising anything. Either way, it's still correct. Factorising just tidies it up a bit.

The question explicitly asked me to factorize it, not expand it

If the question explicitly asks you to factorise it, then you have no other choice but spot all these tricks.

To answer such questions, I would make the computation of the determinant as simple as possible. For example, performing row operations to obtain 1, 0, 0 in the third column. That way, you can ignore the second and third terms.
Then just find the determinant in the same way, bringing out common factors that you notice in each of the rows.

Why do I have no choice xD? Why can't I just expand the matrix and then factorize it; would I not get marks for doing that?

In a 3x3 matrix can't I just expand it out and then factorise it ? It is much faster and I'm less likely to make stupid mistakes.

$[br][br]sinx I cosx I 1[br]sin^2x | cos^2x |1[br]sin^3x | cos^3x |1[br]$
no modulus just trying to make the matrices lines clearer