Given:
$ e^{2jt} \rightarrow SYSTEM \rightarrow te^{-2jt} $
$ e^{-2jt} \rightarrow SYSTEM \rightarrow te^{2jt} $
Solve:
$ cos(2t) \rightarrow SYSTEM \rightarrow ? $
At this point, we must use Euler's relation to expand cos(2t) into exponentials. Then, we will be able to use the given inputs and corresponding outputs to come to a conclusion.
$ cos(2t) = \frac{e^{2jt}+e^{-2jt}}{2} $
$ cos(2t) \rightarrow SYSTEM \rightarrow \frac{1}{2}te^{-2jt}+\frac{1}{2}te^{2jt} $
