That's right, an infinite product for sin(x) courtesy of Euler (feat. a crazy series for cot(x))

[@sarthak.chatterjee1]

Beautiful, one of my favorite results in all of math! A fun fact/observation: Note how the infinite series for cot(x) you derive is exactly of the form that x - cot(x) has the perfect structure for Glasser's Master Theorem (GMT) to hold! So, we have that the integral from -infinity to + infinity of f(x - cot(x)) dx is exactly equal to the integral from -infinity to + infinity of f(x) dx. This fact/technique/trick, combined with a proper choice of f(x) eases the evaluation of certain tricky integrals, e.g., the integral from -infinity to + infinity of 1/(1+( x + tan(x) )^2) dx. Would love to see some videos on applications of the GMT on your channel!

[@maths_505]

Noted

[@Sai404wastaken]

Now that's bonkers

[@truIyepic]

your videos are getting crazier and crazier, and I love it 🔥 🔥

[@maths_505]

Thanks mate although this one was alot more on the academic side but I really love this series for the cotangent. Oh and if you differentiate it you get an awesome expansion for csc²(x)

[@truIyepic]

​@maths_505 personally I find academic problems really interesting, especially as we don't learn this at school yet. And yeah, differentiating does produce a cool expansion, I didn't think of that 😅

[@ShinjiCarlos]

Holly mother of Gosh! This proof of cot z is awesome! I searched for that all over internet and textbooks, but never reached the answer. Thanks

[@gonzus1966]

That was awesome. Euler really is the master of us all.

[@maths_505]

The real top G

[@GiornoYoshikage]

I noticed it's possible to derive series expansion of 'cot(πx)' using properties of digamma function (corollary of the Euler's reflection formula for gamma function & series expansion of digamma function). Seems to be short and beautiful result but requires knowledge about gamma-digamma brothers. My first derivation of the expansion was based on Fourier series too

[@lorck3261]

Why at minute 8:46 did you write "cos(k.x)" instead of "cos(k.t.x)" if the function is evaluated in "z", and "z = t.x". In this case, wouldn't you have to do: "cos(k.t.x)"? It would not be possible to do a manipulation like: "(-1)^(k)" because "t" is not an integer. Please, I really need this answer.

[@MrWael1970]

Amazing Proof. Thank you for your fruitful effort.

[@emanuellandeholm5657]

In order to keep our conversation going, in the light of you nuking that last video: Thanks friend! Your channel has been helping me in trying times. And I hope, and anticipate, that you're going to the moon bro!

[@maths_505]

Bruh I ain't forgetting no one especially the friends I've made. Its cuz of all of you that my channel is doing good alhamdullilah. Hopefully I get to travel the world soon and I really want to meet you and so many others. Nd I'm here whenever you need bro. Keep grinding #hustle #neverstop

[@emanuellandeholm5657]

@@maths_505 Much love bro!

[@maths_505]

Oh and I had to nuke that video cuz it was doing really well despite the error in it so I'll reupload it soon.

[@emanuellandeholm5657]

@@maths_505 It's fine.

[@maths_505]

@@emanuellandeholm5657 SUIIIIIIIIIIIIIIII

[@ShinjiCarlos]

My Gosh, that is jaw dropping. Made my saturday night. Gorgeous!

[@ΙΗΣΟΥΣΧριστος-θ2γ]

Masterpiece video. As a small request, could you nuke some more integrals with contour integration. I dont know about others but i am kind of a sucker for complex analysis. You could also follow up with eulers proof of the basel problem since the sin product form and it are basically synonymous.

[@maths_505]

Oh yes ofcourse I'd love to. Could you do me a favor and DM this to my instagram so I remember

[@cameronspalding9792]

The formula for the Fourier series term only works when t^2 is not k^2

[@banjo2402]

I love the fact that I come here without any knowledge about advanced math (so I don't understand where all the weird functions like zeta, eta and gamma etc. come from) but I can understand the algebra and most of the calculus so I can still enjoy the vids

[@MirzaAli-r5c]

Sin(3)=3 sin(1)-4(sin(1))cubed

[@EtienneSturm1]

that was indeed beautiful. Thanks

[@kkanden]

omg no way i was literally searching for proof of this formula today as i knew that euler didn't actually prove this when researching the zeta function. great video! what device do you use to make your videos?

[@maths_505]

I use my phone

[@kkanden]

@@maths_505 oh do you use a stylus?

[@maths_505]

Yup

[@mikeoffthebox]

It is easy to verify Euler's product against the formula from Fourier analysis just by looking at the logarithmic derivative of sin(x)/x.

[@idjles]

Your final solution is just a polynomial of degree N with roots at every n*pi and then you let N go to infinity. So the result can bee achieved “by inspection” of the sine graph.

[@maths_505]

I believe that's exactly how euler did it before providing a formal proof

[@cristofer6806]

I’d love to see more infinite products

[@MirzaAli-r5c]

You can construct trignometric table if you Have value of sin(1)

[@yoav613]

Nicely done! By the way if you choose x=0.5pi instead of pi,you can derive the series representation of 1/sinx.

[@fartoxedm5638]

I have been delaying the proper proof of this formula for so long so it somehow appeared in front of me by itself! thanks

[@maths_505]

Oh that's awesome mate! And yeah it's a beautiful result. The amazing thing is they extend to complex numbers too cuz complex sin(z) and cot(z) have the same zeros and poles respectively as their real counterparts.

[@MirzaAli-r5c]

Please can you solve Cubic equation Using cardano method

[@maths_505]

Too easy for the channel. I'm sure you can find videos on KZbin relating to that kinda content.

[@ashishraje5712]

Thank u

[@MirzaAli-r5c]

Sin(3)=sin(18-15)

[@bartekabuz855]

Why would you rather write (tsintπ) (-1)^k+1 instead of tsin(tπ) (-1)^k+1? Are you () phobic

[@yoav613]

Very nice!! It seems that fourier series is very powerful tool,if you only know how to choose ther right function!😃💯

[@maths_505]

Yeah I actually got this from a pdf file while searching for a proof of the infinite product

[@yassinezanned9837]

Wow 🤯

[@thomasblackwell9507]

Hate to be rude but isn’t Eular spelled Euler

[@maths_505]

Oh yes you're right dear friend.....just fixed it. Thanks

[@thomasblackwell9507]

@@maths_505 You are welcome. Glad that I could be of some use, so thank you in return.

[@PyarMatKaro]

You told us that x had to be strictly between -pi and pi. Then you set x equal to pi. I don't follow!

[@maths_505]

Oh the fourier series is perfectly valid for the closed interval between -pi and +pi. And in fact we normally use closed intervals whenever possible....I should've written that instead of the open interval even though it's clear that the series works for pi as well.

[@nickruffmath]

V neat 🎉

[@maths_505]

Thanks