Does this system of equations have any solutions? I would just like to know if this system has a solution:
$$\begin{cases}
x = \dfrac{1}{\sqrt{2}} \cos(x+y) \\
y = \dfrac{1}{\sqrt{2}} \sin(x+y)  
\end{cases}$$
I don't know how to do it at all. Could someone help me?

If it did $x^2+y^2=1/2$. On this circle, $t=x+y$ varies between
$-1$ and $1$. So can we find $t$ with $\cos t+\sin t=\sqrt2 t$?
Drawing a graph seems to indicate there's just one such $t$.
