$8.424 \cos x+\sqrt{\sin ^{2} x-2 \sin 2 x+4 \cos ^{2} x}=0$.
## Solution.

$\sqrt{\sin ^{2} x-4 \sin x \cdot \cos x+4 \cos ^{2} x}=-\cos x \Leftrightarrow|2 \cos x-\sin x|=-\cos x \Leftrightarrow$

1) $\left\{\begin{array}{l}\cos x \leq 0, \\ 2 \cos x-\sin x=-\cos x\end{array} \Leftrightarrow\left\{\begin{array}{l}\cos x \leq 0, \\ \operatorname{tg} x=3\end{array} \Leftrightarrow x=\operatorname{arctg} 3+\pi(2 k+1)\right.\right.$.
2) $\left\{\begin{array}{l}\cos x \leq 0, \\ 2 \cos x-\sin x=\cos x\end{array} \Leftrightarrow\left\{\begin{array}{l}\cos x \leq 0, \\ \operatorname{tg} x=1\end{array} \Leftrightarrow x=\frac{\pi}{4}+\pi(2 k+1)\right.\right.$.

Answer: $x_{1}=\operatorname{arctg} 3+\pi(2 k+1), \quad x_{2}=\frac{\pi}{4}+\pi(2 k+1)$.