5. Solve the system of equations $\left\{\begin{array}{l}4 \cos ^{2} x \cdot \sin ^{2} \frac{x}{6}+4 \sin \frac{x}{6}-4 \sin ^{2} x \cdot \sin \frac{x}{6}+1=0, \\ \sin \frac{x}{4}=\sqrt{\cos y} .\end{array}\right.$ (10 points)

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# Solution:

Solve the 1st equation of the system:

$$
\begin{aligned}
& 4 \cos ^{2} x \cdot \sin ^{2} \frac{x}{6}+4 \sin \frac{x}{6}-4 \sin ^{2} x \cdot \sin \frac{x}{6}+1=0 \quad \Leftrightarrow \quad 4 \cos ^{2} x \cdot \sin ^{2} \frac{x}{6}+4 \cos ^{2} x \cdot \sin \frac{x}{6}+1=0 \\
& \Leftrightarrow\left(2 \cos x \sin \frac{x}{6}+\cos x\right)^{2}+\sin ^{2} x=0 \quad \Leftrightarrow\left\{\begin{array} { l } 
{ 2 \cos x \left( \sin \frac { x } { 6 } + \frac { 1 } { 2 } \right) = 0 , } \\
{ \sin x = 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
\sin \frac{x}{6}=-\frac{1}{2}, \\
\sin x=0,
\end{array} \Leftrightarrow\right.\right. \\
& \left\{\begin{array} { l } 
{ [ \begin{array} { l } 
{ x = 6 \left( - \frac { \pi } { 6 } + 2 \pi k \right) , k \in \mathbb{Z} , } \\
{ x = 6 \left( - \frac { 5 \pi } { 6 } + 2 \pi k \right) }
\end{array} ) , k \in \mathbb{Z} , } \\
{ x = \pi n , n \in \mathbb{Z} }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ [ \begin{array} { l } 
{ x = - \pi + 12 \pi k , k \in \mathbb{Z} , } \\
{ x = - 5 \pi + 12 \pi k , k \in \mathbb{Z} , } \\
{ x = \pi n , n \in \mathbb{Z} }
\end{array} }
\end{array} \Leftrightarrow \left[\begin{array}{c}
x=-\pi+12 \pi k, k \in \mathbb{Z} \\
x=-5 \pi+12 \pi k, k \in \mathbb{Z}
\end{array}\right.\right.\right.
\end{aligned}
$$

Returning to the original system of equations, we obtain an equivalent system

![](https://cdn.mathpix.com/cropped/2024_05_06_18575ac96ab43cb21cd9g-04.jpg?height=339&width=1608&top_left_y=1047&top_left_x=287)

Answer: $\left(11 \pi+24 \pi m, \pm \frac{\pi}{3}+2 \pi n\right),\left(-5 \pi+24 \pi m, \pm \frac{\pi}{3}+2 \pi n\right), m, n \in \mathbb{Z}$.