$\begin{array}{l} {\sin ^2}x\left( {1 + \sin x} \right) + {\cos ^2}x\left( {1 + \cos x} \right) = 2{\cos ^2}x\\ \Leftrightarrow {\sin ^3}x + {\sin ^2}x + {\cos ^2}x + {\cos ^3}x - 2{\cos ^2}x = 0\\ \Leftrightarrow \left( {{{\sin }^3}x + {{\cos }^3}x} \right) + {\sin ^2}x - {\cos ^2}x = 0\\ \Leftrightarrow \left( {\sin x + \cos x} \right)\left( {{{\sin }^2}x - \sin x\cos x + {{\cos }^2}x} \right) + \left( {\sin x - \cos x} \right)\left( {\sin x + \cos x} \right) = 0\\ \Leftrightarrow \left( {\sin x + \cos x} \right)\left( {1 + \sin x - \cos x - \sin x\cos x} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l} \sin x + \cos x = 0\\ \cos x - \sin x + \sin x\cos x = 1 \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} \sin x + \cos x = 0\\ \cos x\left( {1 + \sin x} \right) - \left( {\sin x + 1} \right) = 0 \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} \sqrt 2 \sin \left( {x + \dfrac{\pi }{4}} \right) = 0\\ \left( {\cos x - 1} \right)\left( {\sin x + 1} \right) = 0 \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} x + \dfrac{\pi }{4} = k\pi \\ \cos x = 1\\ \sin x = - 1 \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = \dfrac{{ - \pi }}{4} + k\pi \\ x = m2\pi \\ x = - \dfrac{\pi }{2} + l2\pi \end{array} \right.\left( {k,m,l \in \mathbb{Z}} \right) \end{array}$
