$(x+1)^6 + (x-1)^6 = 0 $

$\left ( \dfrac {x+1} {x-1} \right )^6 + 1=0 $

$\left ( \dfrac {x+1} {x-1} \right )^6 =-1 = \cos \pi + i \sin \pi \cdots \{ \because, \cos \pi =-1, \sin \pi =0 \} $

$\left ( \dfrac {x+1} {x-1} \right )^6 = \left ( \cos \left ( \pi +2 n \pi \right )+ i \sin \left ( \pi + 2n \pi \right ) \right )^{\frac {1}{6}}, \ n=0,1,2,3,4,5.$

Using DeMovire's Theorem,

$\left ( \dfrac {x+1}{x-1} \right ) = \left ( \cos \dfrac {(\pi + 2n \pi)}{6}+i \sin \dfrac {(\pi + 2 n \pi)}{6} \right )$

$let \ \dfrac {(\pi + 2n \pi)}{6}=y$

$\left ( \dfrac {x+1}{x-1} \right )=\cos y+ i \sin y$

By componendo - dividendo,

$\dfrac {x+1+x-1}{x+1-x+1} = \dfrac {\cos y + i \sin y+1}{\cos y + i \sin y-1}$

$\dfrac {2x}{2}= \dfrac {2 \cos^2 \left ( \frac {y}{2} \right )+i \left (2 \sin \left ( \frac {y}{2} \right )\cos \left ( \frac {y}{2} \right ) \right )} { -2 \sin ^2 \left ( \frac {y}{2} \right )+ i \left ( 2 \sin \left ( \frac {y}{2} \right ) \cos \left ( \frac {y}{2} \right ) \right )}$

$\{\because, 1-\cos 2A =2 \sin^2 A, 1 + \cos 2A = \cos^2 A, \sin 2A = 2 \sin A \cos a \}$

$x= \dfrac {2\cos \left ( \frac {y}{2} \right ) \left ( \cos \left ( \frac {y}{2} \right )+ i \sin \left (\frac {y}{2} \right ) \right )} { 2 \sin \left ( \frac {y}{2} \right )\left ( -\sin \left ( \frac {y}{2}\right ) + i \cos \left ( \frac {y}{2} \right )\right )}= \cot \left ( \dfrac {y}{2} \right ) \dfrac {\left ( \cos \left ( \frac {y}{2} \right ) + i \sin \left (\frac {y}{2} \right ) \right )} { i \left ( \cos \frac {y}{2} \right )- \frac {1}{i}\sin \left (\frac {y}{2} \right )}$

$x=\left (\dfrac {1}{i} \right )\cot \left ( \dfrac {y}{2} \right ) \dfrac {\left ( \cos \left (\frac {y}{2} \right ) + i \sin \left ( \frac {y}{2} \right )\right )} { \cos \left ( \frac {y}{2} \right )- (-i) \sin \left ( \frac {y}{2} \right )} = -i \cot \left ( \dfrac {y}{2} \right ) \dfrac { \left ( \cot \left (\frac {y}{2} \right )+ i \sin \left (\frac {y}{2} \right ) \right )} { \left ( \cos \left ( \frac {y}{2}\right )+ i \sin \left ( \frac {y}{2} \right ) \right )} $

$\left \{ \because , \dfrac {1}{i}= -i\right \}$

$x=-i \cot \left ( \dfrac {(1+2n)\pi}{2\times 6} \right ) $

$ x=i \cot \left ( \dfrac {2n+1}{12} \right )\pi , \ n=0,1,2,3,4,5$