Question

Tan^3(x) =1/3tan(x) all the solutions

Answer 1

`\bf tan^3(x) = \cfrac{1}{3}tan(x)\implies tan^3(x)-\cfrac{1}{3}tan(x) = 0 \\\\\\ tan(x)\left(tan^2(x)-\cfrac{1}{3} \right)=0 \\\\[-0.35em] ~\dotfill\\\\ tan(x) = 0\implies x = tan^(-1)(x)\implies \boxed{x = n\pi \qquad n\in \mathbb{Z}} \\\\[-0.35em] ~\dotfill\\\\ tan^2(x)-\cfrac{1}{3}=0\implies tan^2(x) = \cfrac{1}{3}\implies tan(x) = \pm \sqrt{\cfrac{1}{3}}`

`\bf tan(x) = \pm \cfrac{1}{√(3)}\implies tan(x) = \pm \cfrac{√(3)}{3}\implies x = tan^(-1)\left( \pm \cfrac{√(3)}{3} \right) \\\\\\ \boxed{x = \pm\cfrac{\pi }{6}n~~,~~\pm \cfrac{5\pi }{6}n~~,~~\pm \cfrac{7\pi }{6}n~~,~~\pm \cfrac{11\pi }{6}n\qquad n\in \mathbb{Z}}`