My edit was rejected. Yet,
$$-\text{tr}\left[\text{adjoint}\left[(P^{-1}DP+I)\sin(\Omega t)^2(I+P^{-1}DP)\right] (P^{-1}DP\cos(\Omega t)-\cos(\Omega t)P^{-1}DP)\right]$$
is much more readable (and in fact... correct) than $-\text{trace}(\text{Adjoint}((P^{-1}DP+I)\sin(\Omega t)^2(I+P^{-1}DP)) (P^{-1}DP\cos(\Omega t)-\cos(\Omega t)P^{-1}DP)$, in my opinion.
Also, my first edit was accepted (did not have time to do all at once) so now the answer looks weird, with both $trace(Adjoint)$ and $\text{tr}(\text{adjoint})$. See the accepted answer.
I am aware that everybody complaining about rejected edits would be problematic, but I'd like to improve my understanding as I'm getting more and more involved on SEs.