Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 2

Multiply numerator and denominator by the denominator's complex conjugate:Thus the real part and imaginary part are

The polar form is $1+i\sqrt{3}=2e^{i\pi /3}$. By De Moivre's theorem,Therefore the real part and imaginary part are

Using Euler's formula, the difference of the two complex exponentials is $2i\sin (\pi /4)=i\sqrt{2}$. HenceThese are the real part and imaginary part, respectively.