I seem to have partially gotten this right, but am missing some answers, so chances are i stumbled across the (partial) right answer improperly. If someone could tell me what I'm doing wrong, that would be grand

on an interval of 0 - 2pi

$\displaystyle tan2x - cotx = 0$

using double angle identities and cotanget identity

$\displaystyle \frac{2tanx}{1-tan^2x} - \frac{1}{tanx} = 0$

$\displaystyle \frac{2tanx}{1-tan^2x} = \frac{1}{tanx}$

multiply both sides by tanx

$\displaystyle \frac{2tan^2x}{1-tan^2x} = 1$

multiply both sides by $\displaystyle 1-tan^2x$

$\displaystyle 2tan^2x = 1-tan^2x$

$\displaystyle 3tan^2x = 1$

$\displaystyle tan^x = \frac{1}{3}$

$\displaystyle tanx = \sqrt{\frac{1}{3}}$

so that gives an answer at $\displaystyle \frac{pi}{6}, \frac{5pi}{6}, \frac{7pi}{6} and \frac{11pi}{6}$, but the book specifies additional answers at $\displaystyle \frac{pi}{2}, \frac{3pi}{2}$. Did I make a mistake above? If not, where are those 2 additional answers from?