I'm guessing the equation is supposed to read
[tex]\tan\left(\dfrac{5\pi}7 x\right) + \sqrt3 = 0[/tex]
Move the constant term to the other side:
[tex]\tan\left(\dfrac{5\pi}7 x\right) = -\sqrt3[/tex]
Take the inverse tangent of both sides:
[tex]\tan^{-1}\left(\tan\left(\dfrac{5\pi}7 x\right)\right) = \tan^{-1}\left(-\sqrt3\right) + n\pi[/tex]
(where n is an integer)
[tex]\dfrac{5\pi}7 x = - \tan^{-1}\left(\sqrt3\right) + n\pi[/tex]
Since tan(π/3) = √3, we get
[tex]\dfrac{5\pi}7 x = -\dfrac\pi3 + n\pi[/tex]
and solving for x gives
[tex]\boxed{x = -\dfrac7{15} + \dfrac{7n}5}[/tex]
