A linear system’s response to $ e^{2jt} $ is $ t*e^{-2jt} $, and its response to $ e^{-2jt} $ is $ t*e^{2jt} $.
What is the response to $ cos(2t) $?
The first thing when solving this problem is to realize the that $ cos(2t) = \frac{e^{2t} + e^{-2t}}{2} = \frac{e^{2t}}{2} + \frac{e^{-2t}}{2} $.
Being a linear system, you can pass each of the two parts shown above individually and add the results together.
Therefore, $ \frac{e^{2t}}{2} \to t*\frac{e^{-2t}}{2} $ and $ \frac{e^{-2t}}{2} \to t*\frac{e^{2t}}{2} $.
Adding the results together yields $ t*\frac{e^{-2t}}{2} + t*\frac{e^{2t}}{2} = t * cos(2t) $
