You can also use `MeshFunctions`:

      Plot[{Cos[x], x Sin[x]}, {x, -3 Pi, 3 Pi}, 
         MeshFunctions -> {(Cos[#] - # Sin[#]) &}, Mesh -> {{0}}, 
         MeshStyle -> Directive[Red, PointSize[Large]]]

![plot of Cos[x] and x Sin[x]][1]

Update: Dealing with `Tan[x]` using `Exclusions` 

      Plot[{Tan[x], x Sin[x]}, {x, -3 Pi, 3 Pi}, 
       MeshFunctions -> {(Tan[#] - # Sin[#]) &}, Mesh -> {{0}}, 
       MeshStyle -> Directive[Red, PointSize[Large]], 
       Exclusions -> Range[-5 Pi/2, 5 Pi/2, Pi]]
       (* or Exclusions -> (Cos[x] == 0) *)

![plot of Tan[x] and x Sin[x]][2]

  [1]: https://i.sstatic.net/W1OJS.png
  [2]: https://i.sstatic.net/8TB1X.png