Animate[Show[ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi},PlotRange->{{-5.1,5.1},{-8.1,8.1}}],Graphics[{Red,Line[{{Cos[x],Sin[x]},{Cos[x]+x*Sin[x],Sin[x]-x*Cos[x]}}]}],ParametricPlot[{Cos[t],Sin[t]},{t,x,2Pi},PlotRange->{{-5.1,5.1},{-8.1,8.1}},PlotStyle->Red],ParametricPlot[{Cos[t]+t*Sin[t],Sin[t]-t*Cos[t]},{t,0,x}, PlotRange->{{-5.1,5.1},{-8.1,8.1}},PlotStyle->Black], Graphics[{Red,PointSize[Medium],Point[{Cos[x],Sin[x]}]}]],{x,0,2Pi}]