Step 1: Factor the pair
$x^2 - (1+\sqrt{3})xy + \sqrt{3}y^2 = (x - y)(x - \sqrt{3}y)$. Check: the $xy$ term is $-(1+\sqrt{3})$ and the $y^2$ term is $\sqrt{3}$.
Step 2: Slopes
From $x - y = 0$ the slope is $1$, so $\tan\alpha = 1$. From $x - \sqrt{3}y = 0$ the slope is $\frac{1}{\sqrt{3}}$, so $\tan\beta = \frac{1}{\sqrt{3}}$. So the angles are $45^\circ$ and $30^\circ$.
Step 3: Combine
$\alpha + \beta = 75^\circ$ and $\tan 75^\circ = 2 + \sqrt{3}$.
Step 4: Match
$\frac{\sqrt{3}+1}{\sqrt{3}-1} = \frac{(\sqrt{3}+1)^2}{2} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3}$, which equals $\tan 75^\circ$. So option (C) is right.
Final Answer:
The angles are 45 and 30 degrees, so tan 75 = 2 + sqrt3. This is option (C).
\[ \boxed{\text{(C) }\frac{\sqrt{3}+1}{\sqrt{3}-1}} \]