Oops, I'm a bit late posting the solutions, but I only remembered at 7 and had to write the solution up.
I'm sorry to say that we didn't have any correct entries for either question, with the exception of faf who answered both problems but didn't provide any proof
I'm going to post a solution to the first question now, and the second question a bit later (since the proof is rather long and I don't have a copy of it yet that I can copy, I'll have to work through it again).
Question 1
Calculate the exact value of cos(72), and hence find the value of cos(36)-cos(72).
Perhaps this was a little misleading. My solution is outlined in the zip'd file which contains a Microsoft Word document with the images in.
I should be posting more problems tomorrow night. Have fun with this until then.
maths challenge 1.doc.zip