The golden mean and pi

The golden mean or golden ratio is a very important number due to its occurrences in mathematics, other sciences and nature. It is intrinsically related with the Fibonacci numbers. The golden mean is

It satisfies the equation:

In the present post I will show how the golden mean is related with π, other very important number in mathematics and sciences.

Recently I wrote about Viète's formula, that gives a formula for π as a limit:

We will use the techniques of the past post in order to show the following identity:

This identity is very beautiful and it can be consider as an application of Viète's formula.

In order to prove it, we start with following identity, due to Euler.

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



Let x=π/5,



Then we have:



Using the identity sin²x+cos²x=1, we compute



and



Let x=2π/5 and substitute it in (1), therefore



which yields



Then



Which is the desired identity.

References:
https://en.wikipedia.org/wiki/Viète%27s_formula

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