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

We have,

             $2(\cos \beta -\cos \alpha )+\cos \alpha \cos \beta =1$

or         $4(\cos \beta -\cos \alpha )+2\cos \alpha \cos \beta =2$

$$\begin{gathered} \Rightarrow 1-\cos \alpha +\cos \beta -\cos \alpha \cos \beta =3+3\cos \alpha -3\cos \beta -3\cos \alpha \cos \beta \\ \begin{array}{l}\Rightarrow (1-\cos \alpha )(1+\cos \beta )=3(1+\cos \alpha )(1-\cos \beta )\\ \Rightarrow \frac{(1-\cos \alpha )}{(1+\cos \alpha )}=\frac{3(1-\cos \beta )}{1+\cos \beta }\end{array} \end{gathered}$$

$\begin{array}{l}\Rightarrow {\tan }^2\frac{\alpha }{2}=3{\tan }^2\frac{\beta }{2}\\ \therefore \tan \frac{\alpha }{2}\pm \sqrt{3}\tan \frac{\beta }{2}=0\end{array}$