Basics of Linearity

Question

A linear system's response to $e^{j2t} \$ is $te^{-j2t} \$ and its response to $e^{-2jt}\$ is $t e^{2jt}\$ .

What is the system's response to $cos(2t)\$ ?

Accord to Euler's formula, $cos(t) = \frac{e^{jt}+e^{-jt}}{2}$

Hence, the response to $cos(2t)\$ would be $\frac{e^{j2t}+e^{-j2t}}{2}\$ .

With that in mind, the response to $e^{j2t}\$ is $t e^{-j2t}\$ and its response to $e^{-j2t}\$ is $t e^{j2t}\$ .

Thus the output of the system will be:

$=t\frac{e^{j2t}+e^{-j2t}}{2}\,$ $=t\cos(2t)\,$