2.11. Consider an LTI system with frequency response $$H(e^{j\omega})=\frac{1-e^{-j2\omega}}{1+\frac12e^{-j4\omega}},\quad-\pi<\omega\le\pi.$$
Determine the output $y[n]$ for all $n$ if the input $x[n]$ for all $n$ is $$x[n]=\sin\left(\frac{\pi n}4\right).$$
This is what I tried so far:
I expressed $x[n]$ in terms of exponentials using Euler's formula.
I then calculated the frequency response to each of the exponentials.
I then multiplied the exponentials by the response.
However now I have a bunch of exponentials and have no idea what to do with them. Could I get some help?