If α ~and& β are the solutions of cot x = - 3 in [0,2π] and α ~and&...

$cot x = - \sqrt{3}$

$⇒ x = \dfrac{5\pi}{6}, \dfrac{11\pi}{6}$ and

$cosec x = - 2$

$⇒ x = \dfrac{7\pi}{6}, \dfrac{11\pi}{6}$

So, $\alpha = \dfrac{11\pi}{6}, \beta = \dfrac{5\pi}{6}, \gamma = \dfrac{7\pi}{6}$

$⇒ \alpha - \beta = \pi$ and $\beta + \gamma = 2\pi$

$⇒ \dfrac{\left|\alpha - \beta\right|}{\beta + \gamma} = \dfrac{\pi}{2\pi} = \dfrac{1}{2}$