Viète formula
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- This article is not about Viète's formulas for symmetric polynomials.
In mathematics, the Viète formula, named after François Viète, is the following infinite product type representation of the mathematical constant π:
The expression on the right hand side has to be understood as a limit expression (as 
)
where an is the nested quadratic radical given by the recursion 
with initial condition 
.
[edit] Proof
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).en:Viète's formula
