Show that in the free oscillations of an LC circuit, the sum of the energies stored in the capacitor and the inductor is constant in time.

At an instant t, the capacitor carries charge q and the current i are given by:
$\begin{array}{r}\mathrm{q}\left(\mathrm{t}\right)={\mathrm{q}}_0\cos \mathrm{\omega t}\\ \mathrm{i}\left(\mathrm{t}\right)=-{\mathrm{q}}_0\mathrm{\omega sin}\mathrm{\omega t}\end{array}$
Energy stored in the capacitor at time t is
${\mathrm{U}}_{\overline{\mathrm{E}}}=\frac{1}{2}{\mathrm{CV}}^2=\frac{1}{2}\frac{{\mathrm{q}}^2}{\mathrm{C}}=\frac{{\mathrm{q}}_0^2}{2\mathrm{C}}{\cos }^2\left(\mathrm{\omega t}\right)$
Energy stored In the inductor at time t is
$\begin{array}{l}{\mathrm{U}}_{\mathrm{M}}=\frac{1}{2}{\mathrm{Li}}^2\\ =\frac{1}{2}{\mathrm{Lq}}_0^2{\mathrm{\omega }}^2{\sin }^2\left(\mathrm{\omega t}\right)\\ =\frac{{\mathrm{q}}_0^2}{2\mathrm{C}}{\sin }^2\left(\mathrm{\omega t}\right)(\because \mathrm{\omega }=1/\sqrt{\mathrm{LC}})\end{array}$
This sum is constant in time as q0 and C, both are time independent.