Continuing from this post –

We start with . Recall the double angle identity for sin: . We will use this a lot.

Multiply our expression by . Then we have

Using the double angle identity, we can reduce this:

So we can rewrite this as

for

Because we know that , we see that $lim_{n \to \infty} \dfrac{\xi / 2^n}{sin(\xi / 2^n)} = 1$. So we see that

Now we set . Also recalling that . What do we get?

This is pretty cool. It’s called Vieta’s Formula for . It’s also one of the oldest infinite products.

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