# Sum to Infinity of Trigonometry to $\pi$

For

$$y=\sum_{n=0}^a2\cdot2^n\cdot\tan\left(\frac{45}{2^n}\right)\cdot\sin\left(\frac{90}{2^n}\right)^2$$

I am currently working on a proof with a good friend of mine that involves adding more and more triangles to the sides of a regular polygon but keeping the longest diagonal constant until eventually, it becomes a circle. And we ended up with this formula.

4-sided regular→ 8-sided regular→ 16-sided regular→ 32-sided regular→... →n-sided regular

(When n tends to infinity, the area will be equal to that of a circle with the longest diagonal as diameter)

We have already tried Geometric Sequence Infinite Sum, but there does not seem to have a common ratio.

Moreover, we have used our calculator to input the numbers up to 128[(sin ...] and we get the value of 3.140... which is very close to π But we can't be completely sure that the infinity sum really equals to π.

That is why we really need your knowledge of Maths to solve this.

Is there a way to prove that when $$a$$ tends to infinity, $$y$$ tends to $$\pi$$?

Thanks in advance and Happy Holidays Everyone! :D

• You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. Commented Dec 25, 2018 at 13:24
• Thanks for the tip :D I will edit this right away!
– user629248
Commented Dec 25, 2018 at 13:25
• Jolly Christmas and Happy Holidays to you, My Good Fellow! :)
– user629248
Commented Dec 25, 2018 at 13:26
• Thank you, and Happy Holidays to you too! Let me know once you've edited the question, then I'll probably upvote :) Commented Dec 25, 2018 at 13:32
• Hey by the way, Shaun, can you tell me how can I type Equations Symbols when I submit my Question?
– user629248
Commented Dec 25, 2018 at 13:43

Hint:

$$2\tan A\cdot\sin^22A=2(4\sin^3A\cos A)=(3\sin A-\sin3A)2\cos A$$ $$=3\sin2A-(\sin4A+\sin2A)=2\sin2A-\sin4A$$

which clearly shows Telescopic form

When I come across things like these: (if you do not need to prove it manually) Wolfram Alpha computation

• Thank you! I have heard of Wolfram before, but is there a manually proof for this question?
– user629248
Commented Dec 25, 2018 at 14:02
• Easy to fool yourself here. If the sum happened to be $\pi + 10^{-16}$ then WA would still say True to your query. Just try this (it's $\pi$ up to 15 digits). In this case it seems it evaluates both sides numerically (to floating point accuracy) and checks if they agree. But it doesn't tell you that this is what it does. Commented Dec 25, 2018 at 14:43
• To check this try to ask WA to just evaluate the sum, it doesn't evaluate it to $\pi$ symbolically. Commented Dec 25, 2018 at 14:44