Solve[t-Sin[t]==x,t]

Solve[t Log[t]==x,t]

Solve[(t^2) Log[t]==x,t]

fung=Table[{t,t-Sin[t]},{t,-2Pi,2Pi,(Pi/200)}]

ListPlot[fung]

funig=Table[{t-Sin[t],t},{t,-2Pi,2Pi,(Pi/200)}]

ListPlot[funig]

Show[ListPlot[funig,PlotStyle->Red],ListPlot[fung,PlotStyle->Blue],AspectRatio->Automatic,ImageSize->200]

ig=Interpolation[funig]

ig[1]

N[ig[1]]

Plot[1-Cos[ig[x]],{x,-2Pi,2Pi},AspectRatio->Automatic]

Show[ParametricPlot[{Cos[t],Sin[t]},{t,0,2 Pi}],Table[
ParametricPlot[{Cos[t0]-t Sin[t0],Sin[t0]+t Cos[t0]},{t,-10,10},PlotStyle->{{Thickness[0.001],Gray}}],{t0,0,2 Pi,(Pi/(2^6))}],ImageSize->250,PlotRange->{{-5,5},{-5,5}},Axes->False,Frame->True]

{2Cos[s]-Cos[2 s],2 Sin[s]-Sin[2 s]}

ff[s_]:=2Cos[s]-Cos[2 s];gg[s_]:=2 Sin[s]-Sin[2 s]

ff'[s]

gg'[s]

Show[ParametricPlot[{ff[t],gg[t]},{t,0,2 Pi}],Table[
ParametricPlot[{ff[t0]+t ff'[t0],gg[t0]+t gg'[t0]},{t,-130,130},PlotStyle->{{Thickness[0.001],Gray}}],{t0,0,2 Pi,(Pi/(2^7))}],ImageSize->250,PlotRange->{{-8,8},{-8,8}},Axes->False,Frame->True]

Show[ParametricPlot[{ff[t],gg[t]},{t,0,2 Pi}],Table[
ParametricPlot[{ff[t0]-t gg'[t0],gg[t0]+t ff'[t0]},{t,-130,130},PlotStyle->{{Thickness[0.001],Gray}}],{t0,0,2 Pi,(Pi/(2^6))}],ImageSize->450,PlotRange->{{-8,8},{-8,8}},Axes->False,Frame->True]

Show[ParametricPlot[{ff[t],gg[t]},{t,0,2 Pi}],Table[
ParametricPlot[{ff[t0]-t gg'[t0],gg[t0]+t ff'[t0]},{t,0,130},PlotStyle->{{Thickness[0.001],Gray}}],{t0,0, Pi/2,(Pi/(2^6))}],ImageSize->450,PlotRange->{{-8,8},{-8,8}},Axes->False,Frame->True]

Manipulate[Show[ParametricPlot[{ff[t],gg[t]},{t,0,2 Pi}],Table[
ParametricPlot[{ff[t0]-t gg'[t0],gg[t0]+t ff'[t0]},{t,0,130},PlotStyle->{{Thickness[0.001],Gray}}],{t0,0,tt,(Pi/(2^6))}],ImageSize->250,PlotRange->{{-8,8},{-8,8}},Axes->False,Frame->True],{tt,(Pi/128),2 Pi}]