There are several formulae that use the Fibonacci numbers to compute PI.Well-known trigonometrical identity that can be used to prove one of this formulae is:

If we introduce supstitations:

where

We have:

The solution of this equation is:

We just proved the following arctanges relationship:

Now we get for x=1 the next relation:

Here`s another simple formula which Euler wrote about in 1738:

So, we get the next relation between Archimedes` constant PI and the Fibonacci numbers:

which only uses odd-indexed Fibonacci numbers.