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][1]) 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}]

Voilá:

[![enter image description here][2]][2]


 [1]: https://mathematica.stackexchange.com/questions/16262/use-different-colorfunction-for-each-function-plotted
 [2]: https://i.sstatic.net/dLNDR.png