Since
which is never zero when x and y are real and y is not zero, and since
the only factors of sin(x) are those of the form
Furthermore, since at each of the points
The derrivative of sin(x) is equal to plus or minus one, so we may conclude that each
of these factors appears exactly once. In addition to this we know that appart
from x=0, each factor has a corresponding factor of the form
so that the two may be combined in the form
We may conclude that we may write sin(x) as
From the product rule for derrivatives it can be shown that
The only other necessary constraint is that the limit of their product exists. These two constraints are satisfied when
We may then write
or alternately, by using
instead of x, 
Because we know that
we can say that
and that is how one may arrive at this particular infinite product for pi. I should note that it doesn't converge very fast. In fact, it converges as slowly as any inverse-square infinite sum for pi. It required, I believe, about ninety-thousand factors to get eight decimal places.