$\begin{array}{l}
\cos x - 5\cos \dfrac{x}{2} - 4 = 0\\
\Leftrightarrow 2{\cos ^2}\dfrac{x}{2} - 1 - 5\cos x\dfrac{x}{2} - 4 = 0\\
\Leftrightarrow 2{\cos ^2}\dfrac{x}{2} - 5\cos \dfrac{x}{2} - 5 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos \dfrac{x}{2} = \dfrac{{5 + \sqrt {65} }}{4}(L)\\
\cos \dfrac{x}{2} = \dfrac{{5 - \sqrt {65} }}{4}(TM)
\end{array} \right.\\
\Rightarrow \cos \dfrac{x}{2} = \dfrac{{5 - \sqrt {65} }}{4}\\
\Rightarrow \dfrac{x}{2} = \pm \arccos \left( {\dfrac{{5 - \sqrt {65} }}{4}} \right) + k2\pi \\
\Leftrightarrow x = \pm 2\arccos \left( {\dfrac{{5 - \sqrt {65} }}{4}} \right) + k4\pi \left( {k \in \mathbb{Z}} \right)
\end{array}$
