Solve and NSolve attempt to find all solutions, but they are mainly concerned with polynomial or analytical functions. I doubt they work with interpolation functions. Here is an attempt at automation of the graphical part, that works, although slowly, in the Sin case:

fo[x] = Sin[x];
firstEq = Cases[ContourPlot[fo'[x2] == (fo[x1] - fo[x2])/(x1 - x2),
      {x1, 0, 16}, {x2, 0, 8}][[1]] // Normal, _Line, All];
secondEq = Cases[ContourPlot[fo'[x1] == fo'[x2],
      {x1, 0, 16}, {x2, 0, 8}][[1]] // Normal, _Line, All];
Complement[Flatten@Outer[RegionIntersection, firstEq, secondEq],
 {EmptyRegion[2]}]

It converts the ContourPlots of the two equations into Line object, and then finds the RegionIntersection of the couples of lines. After that you can start FindRoot from the points you have found. I don't know if this solution is practical in your situation.

Thanks for explaining your code to me. I used the same code (FindRoot) earlier. However, this only gives one solution. Also, I want to sort of automate the process that is my code should be able to find all common tangent pairs for a general curve. The reason for doing so is I will be looking at many curves (free energy curves) and checking it manually is something I can ill-afford.

Slightly modified the list given in the question. Your solution gives me the following.

Could you please help me understand what is going on in this plot ?