Giải phương trình: sin 2 x + cos 2 x − sin x − cos x + 1 = 0

Ta có:

\( \Leftrightarrow 2\sin x.\cos x + 2{\cos ^2}x - 1 - \sin x - \cos x + 1 = 0\)

\( \Leftrightarrow 2\cos x\left( {\sin x + \cos x} \right) - \left( {\sin x + \cos x} \right) = 0\)

\( \Leftrightarrow \left( {\sin x + \cos x} \right)\left( {2\cos x - 1} \right) = 0 \Leftrightarrow \left[ \begin{array}{l}\sin x + \cos x = 0\\2\cos x - 1 = 0\end{array} \right.\)

\( \Leftrightarrow \left[ \begin{array}{l}\tan x =  - 1\\\cos x = \frac{1}{2}\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x =  - \frac{\pi }{4} + k\pi \\x =  \pm \frac{\pi }{3} + m2\pi \end{array} \right.\,\,\left( {k,m \in \mathbb{Z}} \right)\).