I have the function $$ g(x,y,t)=\frac{(8x^2y^2+f(x,y,t)+\cos(2t))(8x^2y^2(1+(x+y)^2)+(x+y)^2(f(x,y,t)-\cos(t)))}{64x^4y^4(1+(x+y)^2)} $$ with $$ f(x,y,t) = 1+2x^2+2y^2-4xy\cos(2t)-2(x+y)\sin(2t). $$ Is is possible to analytically prove $g(x,y,t)\ge1$? I have numerically checked it for many values of variables (x,y,t). Any suggestion will be helpful.

I have the function $$ g(x,y,t)=\frac{(8x^2y^2+f_+(x,y,t)-\cos(2t))(8x^2y^2(1+(x+y)^2)+(x+y)^2(f_-(x,y,t)-\cos(t))+4xy(x+y)\sin(2t))}{64x^4y^4(1+(x+y)^2)} $$ with $$ f_{\pm}(x,y,t) = 1+2x^2+2y^2\pm4xy\cos(2t)-2(x+y)\sin(2t). $$ Is is possible to analytically prove $g(x,y,t)\ge1$? I have numerically checked it for many values of variables (x,y,t). Any suggestion will be helpful.

Just in case, the MMA code of this function is

fp[x_, y_, t_] = 
  1 + 2 x^2 + 2 y^2 + 4 x y Cos[2 t] - 2(x + y)Sin[2t];
fm[x_, y_, t_] = 
  1 + 2 x^2 + 2 y^2 - 4 x y Cos[2 t] - 2(x + y)Sin[2t];
g[x_, y_, 
   t_] = (8 x^2 y^2 + fp[x, y, t] - 
     Cos[2 t]) (8 x^2 y^2 (1 + (x + y)^2) + (x + y)^2 (fm[x, y, t] - 
         Cos[2t]) + 
      4 (x + y) x y Sin[2 t])/(64 x^4 y^4 (1 + (x + y)^2));