= Solution
In the $e^{i\omega t}$ <phasor> convention, the total series <RL circuit> impedance is $R+i\omega L$. The resistor voltage follows the current, so its <transfer function> relative to the applied voltage is
$$
\frac{\widetilde V_R}{\widetilde V_{\mathrm{in}}}=\frac R{R+i\omega L}.
$$
At the stated condition $R=\omega L$,
$$
\boxed{\frac{\widetilde V_R}{\widetilde V_{\mathrm{in}}}=\frac{1-i}{2}=\frac{e^{-i\pi/4}}{\sqrt2}.}
$$
Thus the resistor-voltage <amplitude> is $\boxed{V_{R,0}=V_{\mathrm{in},0}/\sqrt2}$ and it \b[lags by $45^\circ$]. For a cosine input it is $V_R(t)=V_{\mathrm{in},0}\cos(\omega t-\pi/4)/\sqrt2$.
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