Consider the LTI system with frequency response $$H(e^{j\omega}) = \frac{1-e^{-j2\omega}}{1+\frac{1}{2}e^{-j4 \omega}}, -\pi < \omega < \pi$$ Determine the output $y[n]$ for all $n$ if the input $x[n]$ for all $n$ is $$x[n] = \sin \left(\frac{\pi n}{4}\right)$$
My attempt: $x[n]$ is an eigenfunction of the LTI system, so the output have the form $$y[n]=|H(e^{j \omega})|\sin\left(\frac{\pi n}{4} + arg(H(e^{j \omega})\right)$$ But I dont know how to determine the phase and modulus of the frequency response with this form. For example, I think that $$| 1 - e^{-j 2 \omega} | = \sqrt{(1-\cos(2 \omega))^2 + \sin^2(2 \omega)}$$ Analogous, doing for the denominator, I could not simplificate the result.
The answer:
$$y[n]=2 \sqrt{2} \sin \left( \frac{\pi(n+1)}{4} \right) $$