I will be very grateful for any hints in the proof of this expression? maybe I should do some replacement expressions or something like that? any help would be helpful, thanks!
$$\iint \limits_{|x|+|y|\le 20} \frac{\mathrm dx\,\mathrm dy}{50 - 5\sin^5 (x)+\cos^7 (2y)} \, >16$$
In this answer we bound the given integral $I$ from below by more simple integrals. Denote the integration domain $\{(x,y): |x|+|y|\le 20\}$ by $D$ and the integrated function $\tfrac 1{50 - 5\sin^5 x+\cos^7 2y}$ by $f(x,y)$.
Since $f(x,y)\ge \tfrac 1{56}$ for each $x,y\in D$, we have $I\ge \tfrac {|D|}{56}=14.28\dots$, where $|D|=800$ is the area of the domain $D$.
Since for $|t|<1$ we have $\tfrac{1}{1-t}>1+t$, $I\ge I_1=\tfrac 1{50} \int_D 1+t(x,y) dxdy$, where $t(x,y)=\tfrac 1{10}\sin^5 x-\tfrac 1{50}\cos^7 2y$. Since the domain $D$ is symmetric with respect to $y$-axis and the function $\sin^5 x$ is odd, its summand cancels out in $I_1$, so $I_1=16-\tfrac 1{50^2}I’$, where $$I’=\int_D \cos^7 2y\, dxdy=\int_{-20}^20 2(20-|y|) \cos^7 2y\, dy.$$
Since $|\cos^7 2y|\le 1$ for each $(x,y)\in D$, we have $I’\le |D|$, so $I_1\ge 16-\tfrac {|D|}{50^2}=15.68$. Wolfram Alpha numerical calculation(https://www.wolframalpha.com/input/?i=integrate+2(20-abs(y))(cos(2y)%5E7)+dy+from+%E2%80%9320+to+20) gives $I’\simeq 0.92124$ implying $I_1\simeq 15.99959476$, which is almost the required lower bound.
A bit better bound
$$I\ge \frac{|D|^2}{\int_D \frac{1}{f(x,y)}dxdy}=\frac{|D|^2}{\int_D 50 - 5\sin^5 x+\cos^7 2y\, dxdy}=$$ $$\frac{|D|^2}{50|D|+I’}=\frac{16^2}{16+\tfrac 1{50^2}I’}>16-\tfrac 1{50^2}I’$$
should follow from integral Cauchy-Schwarz inequality.
At last, since for $|t|<1$ we have $\tfrac{1}{1-t}>1+t+t^2+t^3$, $$I\ge I_2=\frac 1{50} \int_D 1+t(x,y)+ t^2(x,y)+t^3(x,y) dxdy.$$ Since $t(x,y)\le \tfrac 1{10}+\tfrac 1{50}=0.12$ for each $x,y$,
$$I_2\ge \frac 1{50} \int_D 1+t(x,y)+ t^2(x,y)-0.12^3 dxdy=$$ $$\frac 1{50} \int_D 1+\frac 1{10}\sin^5 x-\frac 1{50}\cos^7 2y + \left(\frac 1{10}\sin^5 x-\frac 1{50}\cos^7 2y\right)^2 -0.12^3 dxdy =$$ $$\frac 1{50} \int_D 1-\frac 1{50}\cos^7 2y + \frac 1{100}\sin^{10} x+\frac 1{250}\sin^5 x\cos^7 2y+ \frac 1{2500}\cos^{14} 2y -0.12^3 dxdy =$$ $$\frac {|D|}{50}(1-0.12^3)+ \frac 1{50} \int_D -\frac 1{50}\cos^7 2y + \frac 1{100}\sin^{10} x+ \frac 1{2500}\cos^{14} 2y\, dxdy>$$ $$15.97235+\frac 1{5000} \int_D -2\cos^7 2y + \sin^{10} x\, dxdy>16.0119.$$
(the final integral we calculated by Mathcad).