Steady State of a Difference Equation using MATLAB
Given the trig identities you can rewrite your input as 
$$x(t) = 3\cdot \sin(0.25\cdot \pi \cdot t+ \pi/3) - \sin(0.75\cdot \pi \cdot t+ \pi) $$

So it's just two sine wave going in, so it will be two sine waves coming out. You could potentially calculate the value of the transfer function $\omega = 0.25 \pi$ and $\omega = 0.75 \pi$ and manually calculate the output this way.