Quantum Principal Component analysis by Seth Lloyd

I am currently reading the paper quantum principal component analysis from Seth Lloyd’s article Quantum Principal Component Analysis There is the following equation stated.

Suppose that on is presented with n copies of $rho$. A simple trick allows one to apply the unitary transformation $e^{-irho t}$ to any density matrix $sigma$ up to $n$th order in $t$. Note that
begin{align}label{eq1}tag{1}
text{tr}_pe^{-iSDelta t}rhootimessigma e^{iSDelta t} &= left(cos^2Delta tright)sigma + left(sin^2 Delta tright)rho – isinDelta tleft[rho, sigmaright]
&= sigma – iDelta tleft[rho, sigmaright] + mathcal Oleft(Delta t^2right)
end{align}
Here $text{tr}_p$ is the partial trace over the first variable and $S$ is the swap operator. $S$ is a sparse matrix and so $e^{-iSDelta t}$ can be performed efficiently [6-9]. Repeated application of eqref{eq1}

I know from the qiskit website, that we can express $mathrm{e}^{igamma B}$ as $cos(gamma)I + isin(gamma)B $ with $gamma$ being some real number and $B$ is an involutory matrix. Can someone explain me why there is only a single $sigma$ and no $rho$ in $(cos^2triangle t)sigma$ and a single $rho$ and no $sigma$ in $(sin^2triangle t)rho$? Is it because of the partial trace?

Any intuition or approach is welcome. Many thanks in advance.