Bước 1:

ĐK: \(\cos x \ne 0\).

\(1 + {\rm{sin}}x + {\rm{cos}}x + {\rm{tan}}x = 0\) \(\Leftrightarrow \left( {1 + \tan x} \right) + \left( {\sin x + \cos x} \right) = 0\)

\( \Leftrightarrow \left( {1 + \dfrac{{\sin x}}{{\cos x}}} \right) + \left( {\sin x + \cos x} \right) = 0\)

\( \Leftrightarrow \dfrac{{\sin x + \cos x}}{{\cos x}} + \left( {\sin x + \cos x} \right) = 0\)

\( \Leftrightarrow \left( {\sin x + \cos x} \right)\left( {\dfrac{1}{{\cos x}} + 1} \right) = 0\)

\( \Leftrightarrow \left( {\sin x + \cos x} \right).\dfrac{{1 + \cos x}}{{\cos x}} = 0\)

Bước 2:

\( \Leftrightarrow \left[ \begin{array}{l}\sin x + \cos x = 0\\\cos x + 1 = 0\end{array} \right.\) \( \Leftrightarrow \left[ \begin{array}{l}\sin x = - \cos x\\\cos x = - 1\end{array} \right.\)

\( \Leftrightarrow \left[ \begin{array}{l}\dfrac{{\sin x}}{{\cos x}} = - 1\\\cos x = - 1\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}\tan x = - 1\\\cos x = - 1\end{array} \right.\) \( \Leftrightarrow \left[ \begin{array}{l}x = - \dfrac{\pi }{4} + k\pi \\x = \pi + k2\pi \end{array} \right.\left( {TM} \right)\)