n = 5 Pi; startpt = {0, 2}; incr = .1;
pursuit[{x0_, y0_}, time1_] := Module[{}, X[t_] = t; Y[t_] = Sin[t];
   nd = NDSolve[{Thread[{x'[t], y'[t]} == {X[t] - x[t], Y[t] - y[t]}/
         Sqrt[(X[t] - x[t])^2 + (Y[t] - y[t])^2]], x[0] == x0, 
      y[0] == y0}, {x, y}, {t, 0, time1}];
   Table[Evaluate[{x[t], y[t]} /. nd[[1]]], {t, 0, time1}]];
pursuit[startpt, n] ;
newptRadius[pa_, pb_, time_] := 
  Module[{pnew, xnew, ynew, pab, radius, dist},
   dist = EuclideanDistance[pa, pb];
   radius = If[dist >= incr, incr, dist];
   pab = {pb[[1]] - pa[[1]], pb[[2]] - pa[[2]]};
   xnew = 
    If[pab[[1]] == 0, pa[[1]], 
     Sign[pab[[1]]] radius/
        Sqrt[1 + (pa[[2]] - pb[[2]])^2/(pa[[1]] - pb[[1]])^2] + 
      pa[[1]]];
   ynew = 
    If[pab[[1]] == 0, pa[[2]] + Sign[pab[[2]]] radius, 
     pa[[2]] + (xnew - pa[[1]]) #[[2]]/#[[1]] &@pab];
   pnew = {xnew, ynew}];
tableB = Table[{X@m, Y[m]}, {m, 0, n, incr}];
tableA = FoldList[newptRadius[#, tableB[[#2]], #2] &, startpt, 
   Range[Length@tableB]];
tableC = Table[{x[t], y[t]} /. nd[[1]], {t, 0, n, incr}];
li = Flatten[{tableA, tableB, tableC}, 
  1]; pr = {{Min@#, Max@#} &@li[[All, 1]], {Min@#, Max@#} &@
   li[[All, 2]]};
Manipulate[
 Graphics[{PointSize[.02], Blue, 
   Line /@ Thread@{Most@#, Rest@#} &@Take[tableA, m], 
   Point@tableA[[m]], Black, 
   Line /@ Thread@{Most@#, Rest@#} &@Take[tableB, m], 
   Point@tableB[[m]], Red, 
   Line /@ Thread@{Most@#, Rest@#} &@Take[tableC, m], 
   Point@tableC[[m]]}, PlotRange -> pr, Frame -> True, 
  AspectRatio -> .5/GoldenRatio] ,
 {m, 1, Length@tableA - 1, 1, Appearance -> "Open", 
  AnimationRate -> 10}, ControlPlacement -> Top]