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$\left\{\begin{array}{c}x-\mathrm{arctan}y=y-\mathrm{arctan}x\\ {x}^{2}-xy+4{y}^{2}=4\end{array}$
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Camron Herrera
Without loss of generality we may suppose that $x\ge y$. But if $x>y$, what is the sign of $\left(\mathrm{arctan}y-\mathrm{arctan}x\right)$? conclude that $x=y$ and finish the job.