System Response
Based on the Euler Formula, $ \cos(2t)\,= \frac{e^{2jt}+e^{-2jt}}{2}\, $.
We already had the response of $ e^{2jt}\, $ is $ te^{-2jt}\, $ and the response of $ e^{-2jt}\, $ is $ te^{2jt}\, $.
Since the system is a LTI system, we have
Output = Response of $ \frac{e^{2jt}+e^{-2jt}}{2}\, $
= Response of $ \frac{e^{2jt}\, $ + Response of $ \frac{e^{-2jt}\, $
$ =\frac{te^{2jt}+te^{-2jt}}{2}\, $ (derive from given conditions)
$ =t\frac{e^{2jt}+e^{-2jt}}{2}\, $
$ =t\cos(2t)\, $
