How to make these lines smooth when using ContourPlot inside of a region?

Question

Consider a region in the $(R,T)$ plane defined by the conditions $$0 < R\leq \pi,\quad |T|+R<\pi\tag{1}$$

Define $t(R,T)$ to be $$t(R,T)=\dfrac{1}{2}\left(\tan\left(\dfrac{T-R}{2}\right)+\tan\left(\dfrac{T+R}{2}\right)\right)\tag{2}.$$

I want to plot level sets of $t(R,T)$ inside the region defined by conditions (1). To do so I used the following code

t[R_, T_] := 1/2 (Tan[(T - R)/2] + Tan[(T + R)/2])
cond1 = 0 <= R <= Pi && Abs[T] + R < Pi && R \[Element] Reals && T \[Element] Reals
cond2 = Reduce[cond1, {R, T}]
reg1 = ImplicitRegion[cond2, {{R, -\[Infinity], \[Infinity]}, {T, -\[Infinity], \[Infinity]}}]
Show[RegionPlot[reg1, PlotStyle -> None], ContourPlot[t[R, T] == -1, {R, T} \[Element] reg1], ContourPlot[t[R, T] == 1, {R, T} \[Element] reg1]]

The resulting plot, however, does not show smooth curves.

What is the issue here? Why isn't Mathematica plotting these curves smoothly? How can I correct that?

Answer 1

It seems that use Contours and RegionFunction is more effective.

f[r_, t_] := 1/2 (Tan[(t - r)/2] + Tan[(t + r)/2]);
cond1 = 0 <= r <= Pi && Abs[t] + r < Pi;
cond2 = Reduce[cond1, {r, t}];
reg1 = ImplicitRegion[cond1, {r, t}];
Show[RegionPlot[reg1, PlotStyle -> None], 
 ContourPlot[f[r, t], {r, 0, Pi}, {t, -4, 4}, Contours -> {1, -1}, 
  RegionFunction -> Function[{r, t}, cond1 // Evaluate], 
  ContourShading -> None, ContourStyle -> {Red, Blue}]]