Output of LTI (in time and frequency $\omega$ domain) : when input goes through LPF
Since this appears to be homework, I can give you a few generic pointers

LTI systems maintain periodicity: if the input is periodic so is the output and vice versa.
If a signal is finite in length it can't be periodic
If a signal has finite bandwidth, it must be infinite in time.
You can model the input as $x(t)$ as cosine multiplied with a rectangular function.
Multiplication in the time domain is convolution in the frequency domain.