\(\displaystyle∵{{\cos{\omega}}_{{0}}{t}}={\frac{{{e}^{{{i}\omega_{{0}}{t}}}+{e}^{{-{i}\omega_{{0}}{t}}}}}{{{2}}}}\)

\(\displaystyle{t}\Rightarrow{\frac{{{1}}}{{{s}^{{2}}}}}\)

\(\displaystyle{t}{{\cos{\omega}}_{{0}}{t}}{u}{\left({t}\right)}\Rightarrow{\frac{{{1}}}{{{2}}}}{\left[{\frac{{{1}}}{{{\left({s}-{i}\omega_{{0}}{t}\right)}^{{2}}}}}+{\frac{{{1}}}{{{\left({s}+{i}\omega_{{0}}{t}\right)}^{{2}}}}}\right]}\)

\(\displaystyle\therefore{L}{\left\lbrace{t}{{\cos{\omega}}_{{0}}{t}}{u}{\left({t}\right)}\right\rbrace}\Rightarrow{\left[{\frac{{{s}^{{2}}-{\omega_{{0}}^{{2}}}}}{{{\left({s}^{{2}}+{\omega_{{0}}^{{2}}}\right)}^{{2}}}}}\right]}\)