Question

Solve the equation.
tan³ x - 3 tan x = 0

Answer 1

${tan}^{3} x - 3tan \: x = 0 \\ \\ tanx( {tan}^{2} x - 3) = 0 \\ \\ either \: tan \: x = 0 \\ \\ or \\ \\ {tan}^{2} x - 3 = 0$
So,

$tan \: x = 0 \\ \\ x = {tan}^{ - 1} (0) \\ \\ = {tan}^{ - 1} (tan \: 0) \\ \\ x \: = 0$
now

${tan}^{2} x - 3 = 0 \\ \\ {tan}^{2} x = 3 \\ \\ tanx = + - \sqrt{3} \\ \\ so \\ \\ tan \: x = \sqrt{3} \\ \\ x = {tan}^{ - 1} (tan( \frac{\pi}{3} ) \\ \\ x = \frac{\pi}{3} \\ \\ and \\ \\ tanx = - \sqrt{3} \\ \\ x = {tan}^{ - 1} (tan( - \frac{\pi}{3} ) \\ \\ x = - \frac{\pi}{3}$
So values of x from the above equation are

x = 0, π/3,-π/3