linear time-constant causal system I have a linear time-constant causal system with the transfer function:

And I have the insignal 
How do I get the output signal? 
I thought of Laplace transform the insignal and then get Y and inverse-laplace to get y(t). But I don't know how to laplace transform cos(t) without a Heaviside(t) factor.

The transfer function of a dynamical linear system is defined as:
$$G(s) = \frac{Y(s)}{U(s)},$$
where $U(s)$ is the Laplace transform of the input signal. The output signal can therefore  be obtained just as follows:
$$Y(s) = U(s) G(s) = \mathcal{L}_s [\cos{t}] \frac{1}{(s+1)^2},$$ and hence:

\begin{align}
\color{blue}{y(t)} = & \mathcal{L}^{-1} [Y(s)] \\ 
= & \mathcal{L}^{-1}\left[  \frac{s}{1+s^2}   \frac{1}{(s+1)^2} \right] \\
= &\mathcal{L}^{-1} \left[ -\frac{1}{2 (1+s)^2}+\frac{1}{2 \left(1+s^2\right)} \right] \\ 
=  & \color{blue}{\left(-\frac{e^{-t} t}{2} + \frac{1}{2} \sin{t} \right) H(t)},\end{align}

where $H(t)$ is the Heaviside function.
Note that the Laplace transform of $\sin{t}$ and $\cos{t}$ are tabulated and can be easily found in any textbook.
Hope this helps.
Cheers!
