The first proof went something like:
Tan(x) can be represented as a continued fraction. If the argument (x) in the particular continued fraction is non zero and rational the number must be irrational. Tan(Pi/4) = 1 (1 is rational) so the argument to the tan must be irrational or zero, it isn't zero obviously so it must be irrational.
It isn't super intuitive but what are you gonna do. I'm waiting for our resident math genius Bad Wigins to come by with some awesome proof.