Let $$\alpha$$ and $$\beta$$ be nonzero real numbers such that $$2(\cos\beta-\cos\alpha)+\cos\alpha\cos\beta=1$$. Then which of the following is/are true?

A

$$\tan(\frac{\alpha}{2})+\sqrt{3} tan(\frac{\beta}{2})=0$$

B

$$\sqrt{3} \tan(\frac{\alpha}{2})+tan(\frac{\beta}{2})=0$$

C

$$\tan(\frac{\alpha}{2})-\sqrt{3} tan(\frac{\beta}{2})=0$$

D

$$\sqrt{3} \tan(\frac{\alpha}{2})-tan(\frac{\beta}{2})=0$$