How to solve the following system of equations if $x,y>0$?
$\begin{cases} \sqrt{\sin^{2}x+\frac{1}{\sin^{2}x}}+\sqrt{\cos^{2}y+\frac{1}{\cos^{2}y}}=\sqrt{\frac{20y}{x+y}} \newline \sqrt{\sin^{2}y+\frac{1}{\sin^{2}y}}+\sqrt{\cos^{2}x+\frac{1}{\cos^{2}x}}=\sqrt{\frac{20x}{x+y}} \end{cases}$
I tried graphing this with Desmos and this is what I got.
If I square both the equations I would get
$\begin{cases} \sqrt{\sin^{2}x+\frac{1}{\sin^{2}x}}+\sqrt{\cos^{2}y+\frac{1}{\cos^{2}y}}=\sqrt{\frac{20y}{x+y}}|()^2 \newline \sqrt{\sin^{2}y+\frac{1}{\sin^{2}y}}+\sqrt{\cos^{2}x+\frac{1}{\cos^{2}x}}=\sqrt{\frac{20x}{x+y}}|()^2 \end{cases}$
$\begin{cases} \sin^{2}x+\cos^{2}y+\frac{1}{\cos^{2}y}+\frac{1}{\sin^{2}x}+2\sqrt{\sin^{2}x+\frac{1}{\sin^{2}x}}\sqrt{\cos^{2}y+\frac{1}{\cos^{2}y}}=\frac{20y}{x+y}\newline \sin^{2}y+\frac{1}{\sin^{2}y}+\cos^{2}x+\frac{1}{\cos^{2}x}+2\sqrt{\sin^{2}y+\frac{1}{\sin^{2}y}}\sqrt{\cos^{2}x+\frac{1}{\cos^{2}x}}=\frac{20x}{x+y} \end{cases}$
I don't think this will help in solving the system.
What should I do next?
