I have two conditions $\textbf{cond.1}$ and $\textbf{cond.2}$ for two variables $x,y\in(0,2\pi)$
$$ \text{cond.1}:=\qquad f_2<f_1<f_3 $$ $$ \text{cond.2}:=\qquad g_2<g_1<g_3 $$
where the functions $f_i$ and $g_i$, $i=1,2,3$ are as follows $$ f_1=3 \cos (5 x+y)+13 \cos (y) $$ $$ f_2=3 \cos (9 x-y)-2 \cos (7 x+y) $$ $$ f_3=\sin ^2(x) $$ and $$ g_1= 3 \cos (x+5 y)-13 \sin (x) $$ $$ g_2= 2 \cos (x+7 y)-3 \cos (x-9 y) $$ $$ g_3=\cos ^2(y) $$
Question. I plot the conditions $\textbf{cond.1}$ and $\textbf{cond.2}$ and the result is as the picture below. Numerically, I see that the blue area in both plots is the same. I want to make sure about this fact analytically. On the other hand, I see that all the functions $f_i$ and $g_i$ are symmetric with respect to the exchange of $y \to x+\frac{\pi}2$; does this symmetry help to prove that the blue area in both plots is the same?
