$\displaystyle {Consider,\ G\left(s\right)=\ \frac{1}{s^2-2s+2}\ }^{\ }\ \ \\[2ex] \begin{align*} \displaystyle L^{-1}\left[G\left(s\right)\right]&=L^{-1}[\frac{1}{s^2-2s+2}] \\[2ex] \displaystyle &=\ L^{-1}\left[\frac{1}{s^2-2s+1+1\ }\right]\\[2ex] &=L^{-1}[\frac{1}{{\left(s-1\right)}^2+1^2}] \\[2ex] \displaystyle &=e^tL^{-1}\left[\frac{1}{s^2+1}\right] \\[2ex] \displaystyle \{ L^{-1} [F(s-a)] 7&=e^aL^{-1}\left[F(s)\right] \} \\[2ex] \displaystyle &=e^t\sin{t\ }=g\left(t\right)……….(A) \\[2ex] \displaystyle \{ L^{-1}\left[\frac{a}{s^2+a^2}\right]&=\sin{at\ }\} \\[2ex] \end{align*}\\[2ex] $

$\displaystyle L^{-1}\left[e^{-\pi{}s}\ G(s)\right]=H\left(t-\pi{}\right)g(t-\pi{}) \\[2ex] \displaystyle {\{L}^{-1}\left[e^{as}X\left(s\right)\right]=H(t+a)x(t+a)\}\ \\[2ex] \displaystyle \therefore{}L^{-1}\left[\frac{e^{-\pi{}s}}{s^2-2s+2}\right]=H\left(t-\pi{}\right)\ e^{t-\pi{}}\sin{(t-\pi{})}\ \ …from\ (A) \\[4ex] \displaystyle \therefore{}L^{-1}\left[\frac{e^{-\pi{}s}}{s^2-2s+2}\right]=-H\left(t-\pi{}\right)e^{t-\pi{}}\sin{t\ } \\[2ex] $