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I want to combine two plots which are in two regions. For an example, I want to plot following two figures in the same plot.

a = Plot[Cos[x], {x, 0, 60}]
b = Plot[Sin[x], {x, 60, 90}]

Could anyone please tell me how to do this without using Piecewise function?

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a = Plot[Cos[x], {x, 0, 60}, PlotStyle -> Blue];
b = Plot[Sin[x], {x, 60, 90}, PlotStyle -> Red];
Show[a, b, PlotRange -> All]

Is this what you want?

plot

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Could anyone please tell me how to do this without using Piecewise function?

Cheating:

Plot[{ConditionalExpression[Cos[x], x <= 60], ConditionalExpression[Sin[x], x >= 60]},
     {x, 0, 90}, PlotStyle -> {Blue, Red}]
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