$$\eqalign{
& \tan x - \cot x + 1 = 0\, \cr
& DKXD:\,\left\{ \matrix{
\sin x \ne 0 \hfill \cr
\cos x \ne 0 \hfill \cr} \right. \Leftrightarrow \sin 2x \ne 0 \cr
& \Leftrightarrow 2x \ne k\pi \Leftrightarrow x \ne {{k\pi } \over 2}\,\,\left( {k \in Z} \right) \cr
& PT \Leftrightarrow \tan x - {2 \over {\tan x}} + 1 = 0 \cr
& \Leftrightarrow {\tan ^2}x + \tan x - 2 = 0 \cr
& \Leftrightarrow \left[ \matrix{
\tan x = 1 \hfill \cr
\tan x = - 2 \hfill \cr} \right. \Leftrightarrow \left[ \matrix{
x = {\pi \over 4} + k\pi \hfill \cr
x = \arctan \left( { - 2} \right) + k\pi \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr} $$
