plotting – Mathematica More Detailed Plot Around Asymptotes

I’m wanting to export some animations from Mathematica involving animating certain plots with asymptotes. At certain points in time, one of the parameters approaches a value where the function no longer has an asymptote. So it gets really steep. Take the following example:

g1[x_, t_] := Tan[x]*Sin[2 Pi t] + Cot[x]*Cos[2 Pi t];
myplot [t_] := 
  Plot[g1[x, t], {x, -1.5 [Pi], 1.5 [Pi]}, 
   PlotRange -> {{-1.5 [Pi], 1.5 [Pi]}, {-1.5 [Pi], 1.5 [Pi]}}, 
   Axes -> False, AspectRatio -> Automatic, 
   PlotStyle -> Thickness[0.01]];

myplot[0.25/720]
myplot[0.5/720]
myplot[1/720]
myplot[2/720]

Notice how, for the first plots, near the asymptote, the function is not plotted all the way to the edge of the graph; it gets cut off:

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This is because when the argument is near zero, the tangent part of the function is disappearing; when it is exactly at zero, that tangent goes away. Basically, I think the function is becoming so steep there, Mathematica is failing to plot it.

For an animation, this is extremely undesirable since it causes flickering. I understand why the veritcal lines would disappear exactly at zero, since the Tan[x] part goes away completely. That’s why, in my animation, I add a small offset to t, say 1/900, so it never reaches zero exactly. But even when I’m around zero, I still get the problem where the steep part of the function disappears.

Basically, how can I tell Mathematica to plot the function where it is extremely steep, all the way to the edge of the graph?