Show that the average power dissipated per cycle in an $\mathrm{L}-\mathrm{C}-\mathrm{R}$ circuit is given by $\mathrm{P}={\mathrm{V}}_{\mathrm{rms}}\times {\mathrm{I}}_{\mathrm{rms}}\times \cos \mathrm{\varphi }$, where $\mathrm{\varphi }$ is the angle of lag or lead.

Show that the average power dissipated in an $\mathrm{AC}$ circuit is, in general, given by

$\mathrm{P}={\mathrm{V}}_{\mathrm{rms}}\times {\mathrm{I}}_{\mathrm{rms}}\times \mathrm{R}/\mathrm{Z}$

where $\mathrm{R}$ is the resistance of the circuit (defined as real part of the complex impedance) and $\mathrm{Z}$ is the impedance.