4. Solve the system of equations $\left\{\begin{array}{l}2 \sin ^{2} x+2 \sqrt{2} \sin x \sin ^{2} 2 x+\sin ^{2} 2 x=0, \\ \cos x=\cos y .\end{array}\right.$.

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# Solution:

Solve the 1st equation of the system:

$$
\begin{aligned}
& 2 \sin ^{2} x+2 \sqrt{2} \sin x \sin ^{2} 2 x+\sin ^{2} 2 x=0 \Leftrightarrow\left(\sqrt{2} \sin x+\sin ^{2} 2 x\right)^{2}+\sin ^{2} 2 x \cos ^{2} 2 x=0 \Leftrightarrow \\
& \left\{\begin{array} { l } 
{ \sqrt { 2 } \sin x + \sin ^ { 2 } 2 x = 0 , } \\
{ \sin ^ { 2 } 2 x \cos ^ { 2 } 2 x = 0 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ \sin x = 0 , } \\
{ \sin 2 x = 0 , }
\end{array} \text { or } \left\{\begin{array}{l}
\sqrt{2} \sin x+1=0, \\
\cos 2 x=0 .
\end{array}\right.\right.\right. \text { The first system has }
\end{aligned}
$$

solution $x=\pi k, k \in Z$. The second system has solutions $x=(-1)^{k+1} \frac{\pi}{4}+\pi k, k \in Z$.

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

$$
\left[\begin{array}{l}
\left\{\begin{array}{l}
x=2 \pi k, k \in Z, \\
\cos y=1 ;
\end{array}\right. \\
\left\{\begin{array}{l}
x=\pi+2 \pi k, k \in Z, \\
\cos y=-1 ;
\end{array}\right. \\
\left\{\begin{array} { l } 
{ \{ \begin{array} { l } 
{ x = 2 \pi k , k \in Z , } \\
{ x = - \frac { \pi } { 4 } + 2 \pi k , k \in Z , } \\
{ y = 2 \pi n , n \in Z ; }
\end{array} } \\
{ \{ \begin{array} { l } 
{ x = \pi + 2 \pi k , k \in Z , } \\
{ y = \pi + 2 \pi n , n \in Z ; }
\end{array} } \\
{ \{ \begin{array} { l } 
{ x = - \frac { \pi } { 4 } + 2 \pi k , k \in Z , } \\
{ y = \pm \frac { \pi } { 4 } + 2 \pi n , n \in Z ; }
\end{array} } \\
{ \{ \begin{array} { l } 
{ x = - \frac { 3 \pi } { 4 } + 2 \pi k , k \in Z , } \\
{ \cos y = - \sqrt { 2 } / 2 }
\end{array} }
\end{array} \Leftrightarrow \left[\begin{array}{l}
x=-\frac{3 \pi}{4}+2 \pi k, k \in Z \\
y= \pm \frac{3 \pi}{4}+2 \pi n, n \in Z
\end{array}\right.\right.
\end{array}\right.
$$

## Answer:

$(2 \pi k, 2 \pi n), \quad(\pi+2 \pi k, \pi+2 \pi n), \quad\left(-\frac{\pi}{4}+2 \pi k, \pm \frac{\pi}{4}+2 \pi n\right), \quad\left(-\frac{3 \pi}{4}+2 \pi k, \pm \frac{3 \pi}{4}+2 \pi n\right), \quad k, n \in Z$.