Finding steady state response of LTI, dissipative systems
If a sinusoidal signal is applied to an LTI system, the output will be having same frequency as input but will have different phase and amplitude. If the input is \$\cos(wt)\$, then output will be:
$$|H(jw)|\cos\left(wt+\angle H(jw)\right)$$
Where \$H(jw)\$ is the frequency response. \$\angle H(jw)\$ is the argument of \$H(jw)\$. 

In your case, the input is sum of two sinusoids with \$w=1\$. Find the output for each one after evaluating magnitude and phase of \$H(jw)\$ at \$w=1\$ and add them to get the result.