Let's convert to degrees; easier to think about.
[tex] \tan \dfrac{5\pi}{6} = \tan 150^\circ [/tex]
[tex] \tan \dfrac{5\pi}{3} = \tan 300^\circ [/tex]
Angles a multiple of 180 degrees or pi radians apart have the same tangent. These are 150 degrees aka 5pi/6 apart so they don't.
