$8\mathrm{.}4$ Suppose x$($t$)=$ sin $200$ pi$($t$)+2$ sin $400$ pi $($t$)$ and g$($t$)=$ x$($t$)$ sin $400$ pi $($t$).$ If the product g$($t$)($sin $400$ pi $($t$))$ is passed through an ideal lowpass filter with cutoff frequency $400$ pi and passband gain of $2,$ determine the signal obtained at the output of the lowpass filter.