Viète's formula for pi

The number π has many surprising and interesting properties. It is defined as the ratio between the perimeter and the diameter of a circle. There are many interesting identities of this number; in the present post I will introduce the Viète's formula, a very beautiful formula involving π:

I few days ago I wrote about the Wallis product, other beautiful formula that express π as an infinite product of rational numbers.

Viète's formula was the first identity in history to express the number π as a limit, and it was published 1593 by the French mathematician François Viète (1540-1603). Using his formula he calculated π with a precision of nine digits.

Here, I will present Euler's proof of the formula. It requieres basic knowledge of trigonometry and limits.

The formula for the sin of the angle double is:

Iterating it (or using induction) we have:



On the other hand, from the well-known limit expression



we conclude
Therefore



This identity is due to Euler.

We recall the formula of the cosine of the half angle:

Let x=π/2, we have:



Hence



Substituting these expressions into (1), we get



or equivalently



where

References:
https://en.wikipedia.org/wiki/Viète%27s_formula
http://www.theoremoftheday.org/GeometryAndTrigonometry/Viete/TotDViete.pdf

All the formulae of this post were typed by myself in LaTeX.