Answer
See steps Below.
Work Step by Step
The energy supplied by the emf device over one cycle is
$
\begin{aligned}
U_t & =\int_0^T P_{\varepsilon} d t=I \varepsilon_m \int_0^T \sin \left(\omega_d t-\phi\right) \sin \left(\omega_d t\right) d t=I \varepsilon_m \int_0^T\left[\sin \omega_d t \cos \phi-\cos \omega_d t \sin \phi\right] \sin \left(\omega_d t\right) d t \\
& =\frac{T}{2} I \varepsilon_m \cos \phi,
\end{aligned}
$
where we have used
$
\int_0^T \sin ^2\left(\omega_d t\right) d t=\frac{T}{2}, \quad \int_0^T \sin \left(\omega_d t\right) \cos \left(\omega_d t\right) d t=0 .
$
