Giải phương trình:sin²x(1+sinx)+cos²????(1+cos????)=2cos²????

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