<< Graphics`Graphics`

Q  9.3  case 1   Mapping by w=Cos[z]    p.232

w = Cos[z ] /. z  ->  x + I y; u = ComplexExpand[Re[w]]; 
    v = ComplexExpand[Im[w]];

ne0 = 4;
yi = Table[i/ne0, {i, 0, ne0}] Pi /2
ne = Length[yi];
pp = Table[{u, v} /. y ->  yi[[i]], {i, ne}];

N[pp /. x -> 0.1]
N[pp /. x -> -0.1]

zu1 = 
      DisplayTogether[
        ParametricPlot[Evaluate[pp[[1]]], {x, -5, 5}, 
          PlotRange ->  {{-3, 3}, {-3, 3}}, 
          AspectRatio ->  Automatic, 
          PlotStyle ->  Hue[0.9]], 
ParametricPlot[
          Evaluate[pp[[2]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.7]], 
        ParametricPlot[Evaluate[pp[[3]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.5]], 
        ParametricPlot[Evaluate[pp[[4]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.3]], 
ParametricPlot[
          Evaluate[pp[[5]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.3]]]

Q  9.3  case 2    Mapping by w=Cosh[z]  p.232

w = Cosh[z ] /. z  ->  x + I y; 
    u = ComplexExpand[Re[w]];
v = ComplexExpand[Im[w]]

ne0 = 6;
yi = Table[
          i/ne0, {i, 0, ne0}] Pi 
ne = Length[yi];
pp = Table[{u, v} /. y ->  yi[[i]], {i, ne}];

N[pp /. x ->  -5]

N[pp /. x -> 5]

zu2 = 
      DisplayTogether[
        ParametricPlot[Evaluate[pp[[1]]], {x, -5, 5}, 
          PlotRange ->  {{-4, 4}, {-4, 4}}, 
          AspectRatio ->  Automatic, 
          PlotStyle ->  Hue[0.9]], 
ParametricPlot[
          Evaluate[pp[[2]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.7]], 
        ParametricPlot[Evaluate[pp[[3]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.5]], 
        ParametricPlot[Evaluate[pp[[4]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.3]], 
        ParametricPlot[Evaluate[pp[[5]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.5]], 
        ParametricPlot[Evaluate[pp[[6]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.7]], 
        ParametricPlot[Evaluate[pp[[7]]], {x, -5, 5}, 
          PlotStyle ->  Hue[0.9]]]

summary

Show[GraphicsArray[{{zu1, zu2}}]]