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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

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  • $\begingroup$ 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. $\endgroup$
    – Shaun
    Commented Dec 25, 2018 at 13:24
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    $\begingroup$ Thanks for the tip :D I will edit this right away! $\endgroup$
    – user629248
    Commented Dec 25, 2018 at 13:25
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    $\begingroup$ Jolly Christmas and Happy Holidays to you, My Good Fellow! :) $\endgroup$
    – user629248
    Commented Dec 25, 2018 at 13:26
  • $\begingroup$ Thank you, and Happy Holidays to you too! Let me know once you've edited the question, then I'll probably upvote :) $\endgroup$
    – Shaun
    Commented Dec 25, 2018 at 13:32
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    $\begingroup$ Hey by the way, Shaun, can you tell me how can I type Equations Symbols when I submit my Question? $\endgroup$
    – user629248
    Commented Dec 25, 2018 at 13:43

2 Answers 2

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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

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

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  • $\begingroup$ Thank you! I have heard of Wolfram before, but is there a manually proof for this question? $\endgroup$
    – user629248
    Commented Dec 25, 2018 at 14:02
  • $\begingroup$ 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. $\endgroup$
    – Winther
    Commented Dec 25, 2018 at 14:43
  • $\begingroup$ To check this try to ask WA to just evaluate the sum, it doesn't evaluate it to $\pi$ symbolically. $\endgroup$
    – Winther
    Commented Dec 25, 2018 at 14:44

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