A Beautiful Expression for Pi
Today I am going to prove a beautiful and somewhat surprising expression for π. The expression that I will prove is the following:
where the denominator is an infinite product of nested surds.
Step 1 of my proof will establish an important trigonometric identity, which we will use in Step 2 by taking limits and applying the result to π.
Proving an important trigonometric identity
Our expression for π can be derived from the following trigonometric identity:
for all positive integers n and where ⍺ is not an integer multiple of π.
We are going to use classic induction to prove this. Remember that you can prove a result for all positive integers n if and only if you can prove it is true for n = 1 (the induction start), and you can prove that if it is true for n = k then it is true for n = k + 1 (the induction step).
To prove the induction start, let’s use the standard trigonometric identity sin(θ+ϕ) = sinθcosϕ + cosθsinϕ. If θ = ϕ = ⍺/2, we have sin⍺ = 2sin(⍺/2)cos(⍺/2). Dividing both sides by cos(⍺/2) gives the result for n = 1.
Now let’s assume the result is true for n = k. We can use the same standard trigonometric identity to derive the result for n = k + 1, as follows:
Hence the result is true for all positive integers n.
Using this to prove our beautiful expression for π
Now we make the observation that as an angle θ approaches zero, the value of (sinθ)/θ approaches 1. From this, with a little simple manipulation, we can say that
So applying limits to both sides of our trigonometric identity from the last section, we can conclude the following:
Now we are going to use another well known trigonometric identity, which is that cos(θ+ϕ) = cosθcosϕ -sinθsinϕ. Again, with θ = ϕ = ⍺/2, we can say that cos⍺ = cos²(⍺/2)-sin²(⍺/2) =…