Problem: Investigate following equality:
$\displaystyle \frac{2}{\pi }=\frac{\sqrt{2}}{2} \frac{\sqrt{2+\sqrt{2}}}{2} \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \text{...}$
Thanks.
My thorough investigation revealed that this is Vičte's formula. Now what?
You can define a circle to be a regular polygon with an infinite number of sides. We can define $\displaystyle \displaystyle \begin{align*} \pi \end{align*}$ as the circumference of a circle of diameter equal to 1 unit. Therefore, we can get an approximation for $\displaystyle \displaystyle \begin{align*} \pi \end{align*}$ by evaluating the perimeter of said regular polygon.
Some algebraic manipulation of this will give the result you are trying to prove.