c
From conservation of angular momentum we get

\({I}_{1} \omega_{1}+{I}_{2} \omega_{2}=\left({I}_{1}+{I}_{2}\right) \omega\)

\(\omega=\frac{{I}_{1} \omega_{1}+{I}_{2} \omega_{2}}{{I}_{1}+{I}_{2}}\)

\({k}_{{i}}=\frac{1}{2} {I}_{1} \omega_{1}^{2}+\frac{1}{2} {I}_{2} \omega_{2}^{2}\)

\({k}_{{f}}=\frac{1}{2}\left({I}_{1}+{I}_{2}\right) \omega^{2}\)

\({k}_{{i}}-{k}_{{f}}=\frac{1}{2}\left[{I}_{1} \omega_{1}^{2}+{I}_{2} \omega_{2}^{2}-\frac{\left({I}_{1} \omega_{1}+{I}_{2} \omega_{2}\right)^{2}}{{I}_{1}+{I}_{2}}\right]\)

Solving above we get

\({k}_{{i}}-{k}_{{f}}=\frac{1}{2}\left(\frac{{I}_{1} {I}_{2}}{{I}_{1}+{I}_{2}}\right)\left(\omega_{1}-\omega_{2}\right)^{2}\)