c
\( E=25 \sin (1000 t) \)

\( \cos \theta=\frac{1}{\sqrt{2}}\)

LR circuit

\( \text { Initially } \frac{R}{\omega_1 L}=\frac{1}{\tan \theta}=\frac{1}{\tan 45^{\circ}}= \)

\( X_L=\omega_1 L \)

\( \omega_2=2 \omega_1 \text {, given } \)

\( \tan \theta^{\prime}=\frac{\omega_2 L}{R}=\frac{2 \omega_1 L}{R} \)

\( \tan \theta^{\prime}=2 \)

\( \cos \theta^{\prime}=\frac{1}{\sqrt{5}}\)