Given the definition of Linear systems we know the response to $ \alpha x_1(t) + \beta x_2(t) $ is $ \alpha y_1(t)+ \beta y_2(t). $
Consider the system:
$ e^{-2jt}\to F ( e^{-2jt} ) \to te^{2jt} $
From the given system we know
$ x(t)\to F( x(t) )\to tx(-t) $
Euler's formula allows to rewrite $ e^{iy} $ as $ e^{iy}=cos{y}+i*sin{y} $
Using this we deduct that x(t) = cos(2t)= $ \frac{e^{2jt}+e^{-2jt}}{2} $
and when we plug it into our system we see $ \frac{te^{-2jt}+te^{2jt}}{2}=t\frac{e^{-2jt}+e^{2jt}}{2}=tcos(2t) $
Therefore we can confidently say that our system yields the output y(t) = t*cos(2t) when the input
x(t) = cos(2t).
