Adding parallel Impedance 1

When there is an inductor impedance and a resistor impedance in series the total value of the impedance is: -

\$Z_{total} = \sqrt{R^2 + X_L^2}\$

Where, in your example R = 10 ohms and XL is 30 ohms (reactive) making the total impedance 31.623 ohms

Assuming this equation is correct: $$Z = \frac{R}{1+i \omega cR} + i \omega L$$

We multiply the first term's nominator and the denominator by the complex conjugate of the denominator: $$.. = \frac{R(1-i\omega c R)}{(1+i \omega cR)(1-i\omega c R)} + i \omega L$$ and thus getting rid of imaginary stuff in the denominator: $$.. = \frac{R(1-i\omega c R)}{(1+ \omega^2 c^2R^2)} + i \omega L$$ And this thing is easily separated to real and imaginary parts: $$.. = \frac{R}{1+ \omega^2 c^2R^2} + i \left(\omega L - \frac{\omega c R^2}{1+ \omega^2 c^2R^2} \right)$$