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] $\tan \left( \frac { \alpha } { 2 } \right) + \sqrt { 3 } \tan \left( \frac { \beta } { 2 } \right) = 0$
[B] $\sqrt { 3 } \tan \left( \frac { \alpha } { 2 } \right) + \tan \left( \frac { \beta } { 2 } \right) = 0$
[C] $\tan \left( \frac { \alpha } { 2 } \right) - \sqrt { 3 } \tan \left( \frac { \beta } { 2 } \right) = 0$
[D] $\sqrt { 3 } \tan \left( \frac { \alpha } { 2 } \right) - \tan \left( \frac { \beta } { 2 } \right) = 0$