$\begin{array}{l}
{\cos ^3}x + {\cos ^2}x - 4{\cos ^2}\dfrac{x}{2} = 0\\
\Leftrightarrow {\cos ^3}x + {\cos ^2}x - 4.\dfrac{{1 + \cos \left( {2.\dfrac{x}{2}} \right)}}{2} = 0\\
\Leftrightarrow {\cos ^3}x + {\cos ^2}x - 2\left( {1 + \cos x} \right) = 0\\
\Leftrightarrow {\cos ^3}x + {\cos ^2}x - 2\cos x - 2 = 0\\
\Leftrightarrow {\cos ^2}x\left( {\cos x + 1} \right) - 2\left( {\cos x + 1} \right) = 0\\
\Leftrightarrow \left( {\cos x + 1} \right)\left( {{{\cos }^2}x - 2} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = - 1\\
{\cos ^2}x = 2(L)
\end{array} \right. \Rightarrow x = \pi + k2\pi \left( {k \in \mathbb{Z}} \right)
\end{array}$
