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As you say, start by expanding $e^{-iS\Delta t}=\cos(\Delta t)I-i\sin(\Delta t)S$, so you'd be calculating $$ (\cos(\Delta t)I-i\sin(\Delta t)S)\rho\otimes\sigma(\cos(\Delta t)I+i\sin(\Delta t)S). $$ If you multiply out all the terms, then the $\cos^2(\Delta t)$ comes from the two $I$ terms, leaving you with $\rho\otimes\sigma$. If you trace out the first ...

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