## [ Systems of nonlinear algebraic equations ]

Solve the system

$$
\begin{aligned}
& x^{2}+y^{2}=1 \\
& 4 x y\left(2 y^{2}-1\right)=1 .
\end{aligned}
$$
Making the substitution $x=\sin \varphi, y=\cos \varphi$, we obtain the equation $4 \sin \varphi \cos \varphi\left(2 \cos ^{2} \varphi-1\right)=1$, that is, $\sin 4 \varphi=1$. From this, $\varphi=\pi / 8+k \pi / 2$.

## Answer

$( \pm \sin \pi / 8, \pm \cos \pi / 8),( \pm \sin 5 \pi / 8, \pm \cos 5 \pi / 8)$, or

$\left( \pm \frac{\sqrt{2-\sqrt{2}}}{2}, \pm \frac{\sqrt{2+\sqrt{2}}}{2}\right),\left( \pm \frac{\sqrt{2+\sqrt{2}}}{2}, \mp \frac{\sqrt{2-\sqrt{2}}}{2}\right)$

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