Express \(\frac{1+i}{\sqrt{3}+i}\) in the form \(x+i y\), where \(x, y \in \mathbb{R}\). By writing each of \(1+i\) and \(\sqrt{3}+i\) in polar form, deduce that

\[ \cos \frac{\pi}{12}=\frac{\sqrt{3}+1}{2 \sqrt{2}}, \quad \sin \frac{\pi}{12}=\frac{\sqrt{3}-1}{2 \sqrt{2}} . \]