$8.454 \cos \sqrt{x}=\cos x$.
## Solution.

$\cos x-\cos \sqrt{x}=0 \Leftrightarrow \sin \frac{x-\sqrt{x}}{2} \cdot \sin \frac{x+\sqrt{x}}{2}=0 \Leftrightarrow\left[\begin{array}{l}x-\sqrt{x}-2 \pi k=0, \\ x+\sqrt{x}-2 \pi k=0\end{array} \Leftrightarrow\right.$

$\Leftrightarrow\left[\begin{array}{l}\sqrt{x}=\frac{1 \pm \sqrt{1+8 \pi k}}{2}, \\ \sqrt{x}=\frac{-1 \pm \sqrt{1+8 \pi k}}{2}\end{array} \Leftrightarrow\left[\begin{array}{l}\sqrt{x}=\frac{1+\sqrt{1+8 \pi k}}{2}, k=0,1, \ldots, \\ \sqrt{x}=\frac{-1+\sqrt{1+8 \pi k}}{2}, k=0,1, \ldots\end{array} \Leftrightarrow\right.\right.$

$\Leftrightarrow \sqrt{x}=\frac{\sqrt{1+8 \pi k} \pm 1}{2}, k=0,1, \ldots \Leftrightarrow x=\frac{2+8 \pi k \pm 2 \sqrt{1+8 \pi k}}{4}, k=0,1, \ldots$.

Answer: $\quad x=\frac{1 \pm \sqrt{1+8 \pi k}}{2}+2 \pi k, k=0,1, \ldots$.

## $8.455|\sin x+\cos x|=\sqrt{2}$.

Solution. Square both sides:

$1+2 \sin x \cdot \cos x=2 \Leftrightarrow \sin 2 x=1 \Rightarrow 2 x=\frac{\pi}{2}+2 \pi k$.

Answer: $x=\frac{\pi}{4}(4 k+1)$.