Question 1
Solve, for $x,y\in\mathbb R$,
$$\begin{cases}\sqrt{\sin^2x+\dfrac1{\sin^2x}}+\sqrt{\cos^2y+\dfrac1{\cos^2y}}=\sqrt{\dfrac{20y}{x+y}},\\[4pt]\sqrt{\sin^2y+\dfrac1{\sin^2y}}+\sqrt{\cos^2x+\dfrac1{\cos^2x}}=\sqrt{\dfrac{20x}{x+y}}.\end{cases}$$
$$\begin{cases}\sqrt{\sin^2x+\dfrac1{\sin^2x}}+\sqrt{\cos^2y+\dfrac1{\cos^2y}}=\sqrt{\dfrac{20y}{x+y}},\\[4pt]\sqrt{\sin^2y+\dfrac1{\sin^2y}}+\sqrt{\cos^2x+\dfrac1{\cos^2x}}=\sqrt{\dfrac{20x}{x+y}}.\end{cases}$$