Initial angular momentum = $I{\omega _1} + I{\omega _2} $

Let $\omega $ be angular speed of the combined system.

Final angular momentum $=2I$$\omega $

$\therefore $   According to conservation of angular momentum 

$I{\omega _1} + I{\omega _2} = 2I\omega \,\,or\,\,\omega  = \frac{{{\omega _1} + {\omega _2}}}{2}$

Initial rotational kinetic energy 

$E = \frac{1}{2}I\left( {\omega _1^2 + \omega _2^2} \right)$

Final rotational kinetic energy 

$\begin{array}{l}
{E_f} = \frac{1}{2}\left( {2I} \right){\omega ^2} = \frac{1}{2}\left( {2I} \right){\left( {\frac{{{\omega _1} + {\omega _2}}}{2}} \right)^2}\\
\,\,\,\,\,\,\,\, = \frac{1}{4}I{\left( {{\omega _1} + {\omega _2}} \right)^2}\\
\therefore \,Loss\,of\,energy\,\Delta E = {E_i} – {E_f}\\
\,\,\,\,\,\, = \,\frac{1}{2}\left( {\omega _1^2 + \omega _2^2} \right) – \frac{I}{4}\left( {\omega _1^2 + \omega _2^2 + 2{\omega _1}{\omega _2}} \right)\\
\,\,\,\,\,\, = \frac{I}{4}\left[ {\omega _1^2 + \omega _2^2 – 2{\omega _1}{\omega _2}} \right] = \frac{I}{4}{\left( {{\omega _1} – {\omega _2}} \right)^2}
\end{array}$