u=0.9;
f[t_]:=N[ArcTan[Tan[t]-u/Cos[t]]];
pi=N[Pi];
v0=f[0.001];
v1=f[pi/2-0.001];
v2=f[pi/2+0.001];
v3=f[pi-0.001];
v4=f[pi+0.001];
v5=f[3 pi/2-0.001];
v6=f[3 pi/2+0.001];
v7=f[2 pi-0.001];
d1=Round[(v2-v1)/pi];
d2=Round[(v4-v3)/pi];
d3=Round[(v6-v5)/pi];
bild1=ParametricPlot[{t,f[t]},{t,0.01,pi/2 -0.001},
  AxesLabel->{"phi","theta"},DisplayFunction->Identity];
bild2=ParametricPlot[{t,f[t]-d1 pi},{t,pi/2 +0.001,pi-0.001},
  AxesLabel->{"phi","theta"},DisplayFunction->Identity];
bild3=ParametricPlot[{t,f[t]-(d1+d2) pi},{t,pi+0.001,3 pi/2-0.001},
  AxesLabel->{"phi","theta"},DisplayFunction->Identity];
bild4=ParametricPlot[{t,f[t]-(d1+d2+d3) pi},{t,3 pi/2+0.001,2 pi -0.001},
  AxesLabel->{"phi","theta"},DisplayFunction->Identity];
bild=Show[bild1,bild2,bild3,bild4,
DisplayFunction->$DisplayFunction,ImageSize->{400,300}]