LTI system output given input and frequency response
Apply the eigenfunction property of the LTI system after decomposing the sinusoidal input by Euler identity. 

The eigenfunction property of the LTI system states that 

$$ x(t) = e^{j \omega_0 t} \implies y(t) = H(\omega_0) e^{j \omega_0 t} $$

where $H(w)$ is the frequency response of the LTI system.

Euler identity states that :

$$ x(t) = \cos(t) = 0.5 \{ e^{jt} + e^{-jt} \} $$

Then due to system being LTI, you can define the output as:

$$ y(t) = 0.5 \{ H(1) e^{jt} + H(-1) e^{-jt} \} .$$

Since the system impulse response was a real function, its frequency response $H(w)$ will be conjugate symmetric; i.e., $H(w) = H^*(-w)$.

Let's call $H(1) = A e^{j\phi} $ where $A = |H(1)|$ is the magnitude and $\phi = \angle{H(1)}$ is the phase of the frequency response at the frequency $\omega = 1$. 

Hence the output will be :
$$ y(t) = 0.5 \{ H(1) e^{jt} + H^*(1) e^{-jt} \} $$

consider the term $z = H(1)e^{jt}$  as a complex variable, then the second term will be $z^*$, hence the output is:

$$y(t) = 0.5 \{ z + z^* \} = \mathcal{Re} \{ z \} $$

$$y(t) = \mathcal{Re} \{ A e^{j\phi} e^{jt} \} = A \cos(t + \phi) = |H(1)| \cos(t + \angle{H(1)})$$