Determining Existence of Transfer Function H(s) and Fourier Transform H(w) for Non-LTI Systems?
Your system is a modulator.

Since \$\cos(2\pi f_0 t)= \dfrac{e^{\,j2\pi f_0 t} + e^{\,-j2\pi f_0 t}}{2} \$ it follows (\$f_0=1\$):

\begin{align*}
  y(t) = \cos(2\pi t)\, x(t) = \dfrac{e^{\,j2\pi f_0 t} + e^{\,-j2\pi f_0 t}}{2} x(t)
  = \dfrac 1 2 x(t) e^{\,j2\pi f_0 t} + \dfrac 1 2 x(t) e^{\,-j2\pi f_0 t}
\end{align*}

Given the following property of the Fourier transform:

\begin{align*}
  F\{x(t)e^{\,j2\pi f_0 t}\} = X(f-f_0)
\end{align*}

Transforming with Fourier y(t) gives:

\begin{align*}
  Y(f) = \dfrac 1 2 X(f-f_0) + \dfrac 1 2 X(f + f_0) = \dfrac 1 2 X(f-1) + \dfrac 1 2 X(f + 1)
\end{align*}