|MadSci Network: Neuroscience|
Skin temperature drops as a result of smoking because nicotine causes vasoconstriction, the constriction or tightening of blood vessels. Since blood not only circulates oxygen and nutrients but also internally generated heat to the extremeties, one of the consequences of lowering blood-flow is the cooling of the extremeties, among them the skin.
You can find a number of references to the vasoconstrictive properties of nicotine throughout the web. Here are a few:
You say that you "must perfom an experiment to prove this and I am not sure how to go about doing that." That's a difficult question without knowing at what level of school or study you are. One thing you might do is find a smoker and take their skin temperature before and after smoking. You might even show that the timing of their "high" correlates (or doesn't) with the effect on skin temperature [because in theory they are both related to the nicotine in the bloodstream].
I hope that helps. If you need further help, don't hesitate to ask.
Try the links in the MadSci Library for more information on Neuroscience.