Calculation in the phasor domain
Phasors are expressed analytically as complex numbers, therefore in the phasor domain the usual current divider formula is:

\$
I_L = I_g \dfrac{R}{R+jX_L}
\$

Hence, using the modulus operator on both members and using its properties you get:

\$
\left|I_L\right| = \left|I_g \dfrac{R}{R+jX_L}\right| = 
\left|I_g\right| \dfrac{\left|R\right|}{\left|R+jX_L\right|} =
\left|I_g\right| \dfrac{\left|R\right|}{\sqrt{R^2+X_L^2}}=
2 \, mA \dfrac{10}{\sqrt{10^2+10^2}}= 2 \, mA \dfrac 1 {\sqrt 2} = \sqrt 2 \, mA
\$