Question 50
Solve the following system of equations on $\mathbb{R}$
$$
\begin{cases} \sqrt{(\sin x)^2 + \frac{1}{(\sin x)^2}} + \sqrt{(\cos y)^2 + \frac{1}{(\cos y)^2}} = \sqrt{\frac{20y}{x+y}}, \\ \sqrt{(\sin y)^2 + \frac{1}{(\sin y)^2}} + \sqrt{(\cos x)^2 + \frac{1}{(\cos x)^2}} = \sqrt{\frac{20x}{x+y}}. \end{cases}
$$
$$
\begin{cases} \sqrt{(\sin x)^2 + \frac{1}{(\sin x)^2}} + \sqrt{(\cos y)^2 + \frac{1}{(\cos y)^2}} = \sqrt{\frac{20y}{x+y}}, \\ \sqrt{(\sin y)^2 + \frac{1}{(\sin y)^2}} + \sqrt{(\cos x)^2 + \frac{1}{(\cos x)^2}} = \sqrt{\frac{20x}{x+y}}. \end{cases}
$$