Let α and β be non zero real numbers such that 2(cos β - cos α) +...

2\left(cos β - cos α\right) + cos αcos β - 1 = 0 .....(i)

Use cos β = \dfrac{1 - \tan^{2} \dfrac{β}{2}}{1 + \tan^{2} \dfrac{β}{2}}andcos α = \dfrac{1 - \tan^{2} \dfrac{α}{2}}{1 + \tan^{2} \dfrac{α}{2}} in (i)

We get \tan^{2} \left(\dfrac{α}{2}\right) = 3\tan^{2} \left(\dfrac{β}{2}\right) \to \tan \left(\dfrac{α}{2}\right) + \sqrt{3}tan \dfrac{β}{2} = 0ortan \left(\dfrac{α}{2}\right) - \sqrt{3}\tan \left(\dfrac{β}{2}\right) = 0