How do I proceed when the current is Io = 2cos (wt)?
You are right but just keep in mind for solving these kind of problems the solution used is called "Sinusoidal Steady-State Analysis" and you can google it if you're uncertain about how it works. But overall what you did is right except that don't forget the voltage across the resistor is not just 2V (it's not a DC voltage). 2V is in phasor, so you have to convert it back to Time-domain form which is 2cos(wt).
Just like this convert the voltage for capacitor and inductor too.

General format for time domain is:

$$ A=M\cos(\omega t\;+\;\theta) $$
which is converted to the phasor form:
$$ A=M\sphericalangle\theta $$

In order to convert, this is how it's done for the voltage across the resistor:

$$ I_{o}=2\cos(\omega t)\quad mA $$
$$ I_{o}=2\sphericalangle0 \quad mA$$
$$ V_{R}=2\sphericalangle0\;mA\times1k\Omega=2\sphericalangle0\quad (V) $$
$$ \Longrightarrow V_{R}=2\cos(\omega t)\quad (V) $$

Try to do the capacitor and inductor yourself and note that the phase is not zero anymore for those devices.