The expression on the right hand side has to be understood as a limit expression (as )
Using an iterated application of the double-angle formula
for sine one first proves the identity
valid for all positive integers n. Letting x=y/2n and dividing both sides by cos(y/2) yields
Using the double-angle formula sin y=2sin(y/2)cos(y/2) again gives
Substituting y=π gives the identity
It remains to match the factors on the right-hand side of this identity with the terms an. Using the half-angle formula for cosine,
one derives that satisfies the recursion with initial condition . Thus an=bn for all positive integers n.
The Viète formula now follows by taking the limit n → ∞. Note here that
as a consequence of the fact that (this follows from l'Hôpital's rule).