Choose two colors:
{c1,c2}={ColorData[97, 1], ColorData[97, 2]} Define a custom color function:
myColorFunction[x_] := If[Cos[x] > Sin[x], c1,c2] Generate two plots (because you can't specify two different color functions in the same plot) and combine them:
g1 = Plot[{Sin[x], Cos[x]}, {x, 0, 2 Pi}, PlotStyle -> {c1,c2}]; g2 = Plot[Min[Sin[x], Cos[x]], {x, 0, 2 Pi}, PlotStyle -> Thickness[0.015], ColorFunctionScaling -> False, ColorFunction -> (myColorFunction[#1] &), Exclusions -> None]; Show[g2, g1, PlotRange -> {-1,1}] VoiláVoilà:
