Somebody really loved Pythagoras, and
compiled 370 proofs or somesuch.

But then made a bold claim that no
trigonometric proofs are possible.

Loomis, Elisha Scott - 1940 ff
https://files.eric.ed.gov/fulltext/ED037335.pdf

Fact or fiction?

Ok, ChatGPT got smarter:

Apply the angle sum/difference identity for cosine:
cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)

Substitute A = B into the equation:
cos(0) = cos^2(B) + sin^2(B)

Simplify further:
1 = cos^2(B) + sin^2(B)

Does this count?

https://chat.openai.com/share/c1b17dfd-8c4d-4633-9081-8bdc62b338f1

Mild Shock schrieb am Montag, 27. November 2023 um 04:30:23 UTC+1:
> Somebody really loved Pythagoras, and
> compiled 370 proofs or somesuch.
>
> But then made a bold claim that no
> trigonometric proofs are possible.
>
> Loomis, Elisha Scott - 1940 ff
> https://files.eric.ed.gov/fulltext/ED037335.pdf
>
> Fact or fiction?

sin cos is defined as:

sin(phi) = a/c, cos(phi) = b/c
- c = hypotenuse
- a = opposite leg
- b = adjacent leg

So when you obtain, by whatever way of proof:

1 = sin(phi)^2 + cos(phi)^2

You have proved:

1 = a^2/c^2 + b^2/c^2

Now multiply each side by c^2:

c^2 = a^2 + b^2

Fritz Feldhase schrieb am Montag, 27. November 2023 um 07:21:40 UTC+1:
> On Monday, November 27, 2023 at 4:42:31 AM UTC+1, Mild Shock wrote:
> > Ok, ChatGPT got smarter:
> >
> > Apply the angle sum/difference identity for cosine:
> > cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
> >
> > Substitute A = B into the equation:
> > cos(0) = cos^2(B) + sin^2(B)
> ChatGPT started with the two formulas ("implicit definitions"):
>
> tan(A) = a/b and
> tan(B) = b/a.
>
> If a =/= b, you cant just "substitute A = B into the equation" later on in the proof.
>
> Actually, A = B would imply that a = b. So the "proof" would only hold for this special case.
>
> Another thought: Pythagoras even holds for "the degenerative case", where, say, c > 0 but either a = 0 or b = 0.
>
> In this case we just would have a^2 + b^2 = c^2.
>
> But the "formula" "derived" by ChatGPT would not "cover" this case (after all, a/b or b/a is not defined then).
> > Simplify further:
> > 1 = cos^2(B) + sin^2(B)
> >
> > Does this count?

I have seen geometric proofs of:

cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)

What do they assume, I don't remember.

Mild Shock schrieb am Dienstag, 28. November 2023 um 00:25:08 UTC+1:
> sin cos is defined as:
>
> sin(phi) = a/c, cos(phi) = b/c
> - c = hypotenuse
> - a = opposite leg
> - b = adjacent leg
>
> So when you obtain, by whatever way of proof:
>
> 1 = sin(phi)^2 + cos(phi)^2
>
> You have proved:
>
> 1 = a^2/c^2 + b^2/c^2
>
> Now multiply each side by c^2:
>
> c^2 = a^2 + b^2
> Fritz Feldhase schrieb am Montag, 27. November 2023 um 07:21:40 UTC+1:
> > On Monday, November 27, 2023 at 4:42:31 AM UTC+1, Mild Shock wrote:
> > > Ok, ChatGPT got smarter:
> > >
> > > Apply the angle sum/difference identity for cosine:
> > > cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
> > >
> > > Substitute A = B into the equation:
> > > cos(0) = cos^2(B) + sin^2(B)
> > ChatGPT started with the two formulas ("implicit definitions"):
> >
> > tan(A) = a/b and
> > tan(B) = b/a.
> >
> > If a =/= b, you cant just "substitute A = B into the equation" later on in the proof.
> >
> > Actually, A = B would imply that a = b. So the "proof" would only hold for this special case.
> >
> > Another thought: Pythagoras even holds for "the degenerative case", where, say, c > 0 but either a = 0 or b = 0.
> >
> > In this case we just would have a^2 + b^2 = c^2.
> >
> > But the "formula" "derived" by ChatGPT would not "cover" this case (after all, a/b or b/a is not defined then).
> > > Simplify further:
> > > 1 = cos^2(B) + sin^2(B)
> > >
> > > Does this count?

You can arrive at the cosinus law using Eulers formula:

Start with Euler's formula:
e^(i(x + y)) = cos(x + y) + i sin(x + y)

Apply the property e^(i(x + y)) = e^(ix) * e^(iy):
cos(x + y) + i sin(x + y) = (cos(x) + i sin(x)) * (cos(y) + i sin(y))

Expand the right side:
cos(x + y) + i sin(x + y) = cos(x)cos(y) + i cos(x)sin(y) + i sin(x)cos(y) + i^2 sin(x)sin(y)

Combine like terms and use i^2 = -1:
cos(x + y) + i sin(x + y) = (cos(x)cos(y) - sin(x)sin(y)) + i (cos(x)sin(y) + sin(x)cos(y))

Now, by comparing the real and imaginary parts on both sides, we get:
cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
sin(x + y) = cos(x)sin(y) + sin(x)cos(y)

Thanks again ChatGPT.
https://chat.openai.com/share/c1b17dfd-8c4d-4633-9081-8bdc62b338f1

So Pythagoras Law is burried in Complex Analysis, can be derived from?

Mild Shock schrieb am Dienstag, 28. November 2023 um 00:33:59 UTC+1:
> I have seen geometric proofs of:
>
> cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)
>
> What do they assume, I don't remember.
> Mild Shock schrieb am Dienstag, 28. November 2023 um 00:25:08 UTC+1:
> > sin cos is defined as:
> >
> > sin(phi) = a/c, cos(phi) = b/c
> > - c = hypotenuse
> > - a = opposite leg
> > - b = adjacent leg
> >
> > So when you obtain, by whatever way of proof:
> >
> > 1 = sin(phi)^2 + cos(phi)^2
> >
> > You have proved:
> >
> > 1 = a^2/c^2 + b^2/c^2
> >
> > Now multiply each side by c^2:
> >
> > c^2 = a^2 + b^2
> > Fritz Feldhase schrieb am Montag, 27. November 2023 um 07:21:40 UTC+1:
> > > On Monday, November 27, 2023 at 4:42:31 AM UTC+1, Mild Shock wrote:
> > > > Ok, ChatGPT got smarter:
> > > >
> > > > Apply the angle sum/difference identity for cosine:
> > > > cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
> > > >
> > > > Substitute A = B into the equation:
> > > > cos(0) = cos^2(B) + sin^2(B)
> > > ChatGPT started with the two formulas ("implicit definitions"):
> > >
> > > tan(A) = a/b and
> > > tan(B) = b/a.
> > >
> > > If a =/= b, you cant just "substitute A = B into the equation" later on in the proof.
> > >
> > > Actually, A = B would imply that a = b. So the "proof" would only hold for this special case.
> > >
> > > Another thought: Pythagoras even holds for "the degenerative case", where, say, c > 0 but either a = 0 or b = 0.
> > >
> > > In this case we just would have a^2 + b^2 = c^2.
> > >
> > > But the "formula" "derived" by ChatGPT would not "cover" this case (after all, a/b or b/a is not defined then).
> > > > Simplify further:
> > > > 1 = cos^2(B) + sin^2(B)
> > > >
> > > > Does this count?

I am already on the next page. How to derive
the subtraction formulas . See my other posts in this thread,
where I found a derivation from Eulers formula.

Credits for the topic go to my daily dose of 9gag.com :

Two 18 years old students just discovered a proof of
Pythagoras that mathematicians said was impossible.
https://9gag.com/gag/aEq4BeO

C-K Shene also arrives at its very last paragraph
at exponentiation somehow. But he seems to be
blind to Eulers formula. What I now wonder whether

something in Complex Analysis, like a simple i^2 = 1,
basically says Pythagoras Formula?

Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 00:59:59 UTC+1:
> On Tuesday, November 28, 2023 at 12:33:59 AM UTC+1, Mild Shock wrote:
> > I have seen geometric proofs of:
> >
> > cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)
> >
> > What do they assume, I don't remember.
> > Mild Shock schrieb am Dienstag, 28. November 2023 um 00:25:08 UTC+1:
> > > sin cos is defined as:
> > >
> > > sin(phi) = a/c, cos(phi) = b/c
> > > - c = hypotenuse
> > > - a = opposite leg
> > > - b = adjacent leg
> > >
> > > So when you obtain, by whatever way of proof:
> > >
> > > 1 = sin(phi)^2 + cos(phi)^2
> > >
> > > You have proved:
> > >
> > > 1 = a^2/c^2 + b^2/c^2
> > >
> > > Now multiply each side by c^2:
> > >
> > > c^2 = a^2 + b^2
> Yes.
>
> Have you seen this approach. Seem to be correct:
> https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf
>
> Mentioned here (have a look!):
> https://www.cut-the-knot.org/pythagoras/TrigProof.shtml
>
> Finally:
> https://www.researchgate.net/publication/374542705_A_New_Approach_to_Proving_the_Pythagorean_Theorem_C-K_Shene

Ok, ChatGPT wants me to deleve the link and create a
new link, so that everything is seen. I am not using ChatGPTs
first proof. Please scroll down, and read what I did.

ChatGPT was only my secretary, my intelligent typist:

- The first thing I obtained was, like in Jason Zimba,
same idea to use cos sum formula:

Apply the angle sum/difference identity for cosine:

cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
Substitute A = B into the equation:
cos(0) = cos^2(B) + sin^2(B)
Simplify further:
1 = cos^2(B) + sin^2(B)

- The second thing I obtained was, I tried to derive the
cos sum formula itself, hinted by C-K_Shene:

Start with Euler's formula:
e^(i(x + y)) = cos(x + y) + i sin(x + y)
Apply the property e^(i(x + y)) = e^(ix) * e^(iy):
cos(x + y) + i sin(x + y) = (cos(x) + i sin(x)) * (cos(y) + i sin(y))
Expand the right side:
cos(x + y) + i sin(x + y) = cos(x)cos(y) + i cos(x)sin(y) + i sin(x)cos(y) + i^2 sin(x)sin(y)
Combine like terms and use i^2 = -1:
cos(x + y) + i sin(x + y) = (cos(x)cos(y) - sin(x)sin(y)) + i (cos(x)sin(y) + sin(x)cos(y))
Now, by comparing the real and imaginary parts on both sides, we get:
cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
sin(x + y) = cos(x)sin(y) + sin(x)cos(y)

https://chat.openai.com/share/7ac61120-0b20-4682-901b-e58f6eb66361

I only used ChatGPT as secretary, didn't rely on some
ideas of ChatGPT. The proofs are mine, since ChatGPT
always needed several attempts, and I guided ChatGPT.

Possibly the best use case for ChatGPT, use it as secretary.

Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:03:47 UTC+1:
> On Tuesday, November 28, 2023 at 12:57:38 AM UTC+1, Mild Shock wrote:
>
> > Thanks again ChatGPT.
> > https://chat.openai.com/share/c1b17dfd-8c4d-4633-9081-8bdc62b338f1
> I already told you that this "proof" is bogus (and why).

Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:12:55 UTC+1:
> On Tuesday, November 28, 2023 at 1:08:33 AM UTC+1, Mild Shock wrote:
>
> > Two 18 years old students just discovered a proof of
> > Pythagoras that mathematicians said was impossible.
> > https://9gag.com/gag/aEq4BeO
> Did they already publish it in a peer reviewed journal?

I don't know the content of the proof of the
"Two 18 years old students from New Orleans"

Do you have a link? The 9gag.com post has their names.

Mild Shock schrieb am Dienstag, 28. November 2023 um 01:16:14 UTC+1:
> Ok, ChatGPT wants me to deleve the link and create a
> new link, so that everything is seen. I am not using ChatGPTs
> first proof. Please scroll down, and read what I did.
>
> ChatGPT was only my secretary, my intelligent typist:
>
> - The first thing I obtained was, like in Jason Zimba,
> same idea to use cos sum formula:
> Apply the angle sum/difference identity for cosine:
>
> cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
> Substitute A = B into the equation:
> cos(0) = cos^2(B) + sin^2(B)
> Simplify further:
> 1 = cos^2(B) + sin^2(B)
> - The second thing I obtained was, I tried to derive the
> cos sum formula itself, hinted by C-K_Shene:
> Start with Euler's formula:
> e^(i(x + y)) = cos(x + y) + i sin(x + y)
> Apply the property e^(i(x + y)) = e^(ix) * e^(iy):
> cos(x + y) + i sin(x + y) = (cos(x) + i sin(x)) * (cos(y) + i sin(y))
> Expand the right side:
> cos(x + y) + i sin(x + y) = cos(x)cos(y) + i cos(x)sin(y) + i sin(x)cos(y) + i^2 sin(x)sin(y)
> Combine like terms and use i^2 = -1:
> cos(x + y) + i sin(x + y) = (cos(x)cos(y) - sin(x)sin(y)) + i (cos(x)sin(y) + sin(x)cos(y))
> Now, by comparing the real and imaginary parts on both sides, we get:
> cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
> sin(x + y) = cos(x)sin(y) + sin(x)cos(y)
> https://chat.openai.com/share/7ac61120-0b20-4682-901b-e58f6eb66361
>
> I only used ChatGPT as secretary, didn't rely on some
> ideas of ChatGPT. The proofs are mine, since ChatGPT
> always needed several attempts, and I guided ChatGPT.
>
> Possibly the best use case for ChatGPT, use it as secretary.
> Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:03:47 UTC+1:
> > On Tuesday, November 28, 2023 at 12:57:38 AM UTC+1, Mild Shock wrote:
> >
> > > Thanks again ChatGPT.
> > > https://chat.openai.com/share/c1b17dfd-8c4d-4633-9081-8bdc62b338f1
> > I already told you that this "proof" is bogus (and why).
> Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:12:55 UTC+1:
> > On Tuesday, November 28, 2023 at 1:08:33 AM UTC+1, Mild Shock wrote:
> >
> > > Two 18 years old students just discovered a proof of
> > > Pythagoras that mathematicians said was impossible.
> > > https://9gag.com/gag/aEq4BeO
> > Did they already publish it in a peer reviewed journal?

Again, I repeat, I didn't use ChatGPTs first proof!
I even had to correct parts of it, you find it in the log:

> Can you shorten the proof, I guess step 10, 11, 12, 13 are
enough. Using the difference of angles formula for cosinus.
Which in your display was wrongly stated. In step 10 there is a plus already.

So ChatGPT doesn't only need several attempts. One
has to be careful with what ChatGPT produces, its can
have bugs, its still full of halucinations. Maybe I should

review the Euler Formula steps that ChatGPT produced.
But glossing over them, they look ok! Right?

Mild Shock schrieb am Dienstag, 28. November 2023 um 01:18:00 UTC+1:
> I don't know the content of the proof of the
> "Two 18 years old students from New Orleans"
>
> Do you have a link? The 9gag.com post has their names.
> Mild Shock schrieb am Dienstag, 28. November 2023 um 01:16:14 UTC+1:
> > Ok, ChatGPT wants me to deleve the link and create a
> > new link, so that everything is seen. I am not using ChatGPTs
> > first proof. Please scroll down, and read what I did.
> >
> > ChatGPT was only my secretary, my intelligent typist:
> >
> > - The first thing I obtained was, like in Jason Zimba,
> > same idea to use cos sum formula:
> > Apply the angle sum/difference identity for cosine:
> >
> > cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
> > Substitute A = B into the equation:
> > cos(0) = cos^2(B) + sin^2(B)
> > Simplify further:
> > 1 = cos^2(B) + sin^2(B)
> > - The second thing I obtained was, I tried to derive the
> > cos sum formula itself, hinted by C-K_Shene:
> > Start with Euler's formula:
> > e^(i(x + y)) = cos(x + y) + i sin(x + y)
> > Apply the property e^(i(x + y)) = e^(ix) * e^(iy):
> > cos(x + y) + i sin(x + y) = (cos(x) + i sin(x)) * (cos(y) + i sin(y))
> > Expand the right side:
> > cos(x + y) + i sin(x + y) = cos(x)cos(y) + i cos(x)sin(y) + i sin(x)cos(y) + i^2 sin(x)sin(y)
> > Combine like terms and use i^2 = -1:
> > cos(x + y) + i sin(x + y) = (cos(x)cos(y) - sin(x)sin(y)) + i (cos(x)sin(y) + sin(x)cos(y))
> > Now, by comparing the real and imaginary parts on both sides, we get:
> > cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
> > sin(x + y) = cos(x)sin(y) + sin(x)cos(y)
> > https://chat.openai.com/share/7ac61120-0b20-4682-901b-e58f6eb66361
> >
> > I only used ChatGPT as secretary, didn't rely on some
> > ideas of ChatGPT. The proofs are mine, since ChatGPT
> > always needed several attempts, and I guided ChatGPT.
> >
> > Possibly the best use case for ChatGPT, use it as secretary.
> > Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:03:47 UTC+1:
> > > On Tuesday, November 28, 2023 at 12:57:38 AM UTC+1, Mild Shock wrote:
> > >
> > > > Thanks again ChatGPT.
> > > > https://chat.openai.com/share/c1b17dfd-8c4d-4633-9081-8bdc62b338f1
> > > I already told you that this "proof" is bogus (and why).
> > Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:12:55 UTC+1:
> > > On Tuesday, November 28, 2023 at 1:08:33 AM UTC+1, Mild Shock wrote:
> > >
> > > > Two 18 years old students just discovered a proof of
> > > > Pythagoras that mathematicians said was impossible.
> > > > https://9gag.com/gag/aEq4BeO
> > > Did they already publish it in a peer reviewed journal?

So no need for that sinking feeling, humans don't need
synapses anymore, AGI has already been archived,

bla bla bla bla. Or does ChatGPT on purpose
make mistakes? So that nobody gets worried?

LoL

Mild Shock schrieb am Dienstag, 28. November 2023 um 01:23:12 UTC+1:
> Again, I repeat, I didn't use ChatGPTs first proof!
> I even had to correct parts of it, you find it in the log:
>
> > Can you shorten the proof, I guess step 10, 11, 12, 13 are
> enough. Using the difference of angles formula for cosinus.
> Which in your display was wrongly stated. In step 10 there is a plus already.
>
> So ChatGPT doesn't only need several attempts. One
> has to be careful with what ChatGPT produces, its can
> have bugs, its still full of halucinations. Maybe I should
>
> review the Euler Formula steps that ChatGPT produced.
> But glossing over them, they look ok! Right?
> Mild Shock schrieb am Dienstag, 28. November 2023 um 01:18:00 UTC+1:
> > I don't know the content of the proof of the
> > "Two 18 years old students from New Orleans"
> >
> > Do you have a link? The 9gag.com post has their names.
> > Mild Shock schrieb am Dienstag, 28. November 2023 um 01:16:14 UTC+1:
> > > Ok, ChatGPT wants me to deleve the link and create a
> > > new link, so that everything is seen. I am not using ChatGPTs
> > > first proof. Please scroll down, and read what I did.
> > >
> > > ChatGPT was only my secretary, my intelligent typist:
> > >
> > > - The first thing I obtained was, like in Jason Zimba,
> > > same idea to use cos sum formula:
> > > Apply the angle sum/difference identity for cosine:
> > >
> > > cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
> > > Substitute A = B into the equation:
> > > cos(0) = cos^2(B) + sin^2(B)
> > > Simplify further:
> > > 1 = cos^2(B) + sin^2(B)
> > > - The second thing I obtained was, I tried to derive the
> > > cos sum formula itself, hinted by C-K_Shene:
> > > Start with Euler's formula:
> > > e^(i(x + y)) = cos(x + y) + i sin(x + y)
> > > Apply the property e^(i(x + y)) = e^(ix) * e^(iy):
> > > cos(x + y) + i sin(x + y) = (cos(x) + i sin(x)) * (cos(y) + i sin(y))
> > > Expand the right side:
> > > cos(x + y) + i sin(x + y) = cos(x)cos(y) + i cos(x)sin(y) + i sin(x)cos(y) + i^2 sin(x)sin(y)
> > > Combine like terms and use i^2 = -1:
> > > cos(x + y) + i sin(x + y) = (cos(x)cos(y) - sin(x)sin(y)) + i (cos(x)sin(y) + sin(x)cos(y))
> > > Now, by comparing the real and imaginary parts on both sides, we get:
> > > cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
> > > sin(x + y) = cos(x)sin(y) + sin(x)cos(y)
> > > https://chat.openai.com/share/7ac61120-0b20-4682-901b-e58f6eb66361
> > >
> > > I only used ChatGPT as secretary, didn't rely on some
> > > ideas of ChatGPT. The proofs are mine, since ChatGPT
> > > always needed several attempts, and I guided ChatGPT.
> > >
> > > Possibly the best use case for ChatGPT, use it as secretary.
> > > Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:03:47 UTC+1:
> > > > On Tuesday, November 28, 2023 at 12:57:38 AM UTC+1, Mild Shock wrote:
> > > >
> > > > > Thanks again ChatGPT.
> > > > > https://chat.openai.com/share/c1b17dfd-8c4d-4633-9081-8bdc62b338f1
> > > > I already told you that this "proof" is bogus (and why).
> > > Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:12:55 UTC+1:
> > > > On Tuesday, November 28, 2023 at 1:08:33 AM UTC+1, Mild Shock wrote:
> > > >
> > > > > Two 18 years old students just discovered a proof of
> > > > > Pythagoras that mathematicians said was impossible.
> > > > > https://9gag.com/gag/aEq4BeO
> > > > Did they already publish it in a peer reviewed journal?

The two girls made it into scientific american:
https://www.scientificamerican.com/article/2-high-school-students-prove-pythagorean-theorem-heres-what-that-means/

But this was 10 days after April Fools day?

Mild Shock schrieb am Dienstag, 28. November 2023 um 01:26:14 UTC+1:
> So no need for that sinking feeling, humans don't need
> synapses anymore, AGI has already been archived,
>
> bla bla bla bla. Or does ChatGPT on purpose
> make mistakes? So that nobody gets worried?
>
> LoL
> Mild Shock schrieb am Dienstag, 28. November 2023 um 01:23:12 UTC+1:
> > Again, I repeat, I didn't use ChatGPTs first proof!
> > I even had to correct parts of it, you find it in the log:
> >
> > > Can you shorten the proof, I guess step 10, 11, 12, 13 are
> > enough. Using the difference of angles formula for cosinus.
> > Which in your display was wrongly stated. In step 10 there is a plus already.
> >
> > So ChatGPT doesn't only need several attempts. One
> > has to be careful with what ChatGPT produces, its can
> > have bugs, its still full of halucinations. Maybe I should
> >
> > review the Euler Formula steps that ChatGPT produced.
> > But glossing over them, they look ok! Right?
> > Mild Shock schrieb am Dienstag, 28. November 2023 um 01:18:00 UTC+1:
> > > I don't know the content of the proof of the
> > > "Two 18 years old students from New Orleans"
> > >
> > > Do you have a link? The 9gag.com post has their names.
> > > Mild Shock schrieb am Dienstag, 28. November 2023 um 01:16:14 UTC+1:
> > > > Ok, ChatGPT wants me to deleve the link and create a
> > > > new link, so that everything is seen. I am not using ChatGPTs
> > > > first proof. Please scroll down, and read what I did.
> > > >
> > > > ChatGPT was only my secretary, my intelligent typist:
> > > >
> > > > - The first thing I obtained was, like in Jason Zimba,
> > > > same idea to use cos sum formula:
> > > > Apply the angle sum/difference identity for cosine:
> > > >
> > > > cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
> > > > Substitute A = B into the equation:
> > > > cos(0) = cos^2(B) + sin^2(B)
> > > > Simplify further:
> > > > 1 = cos^2(B) + sin^2(B)
> > > > - The second thing I obtained was, I tried to derive the
> > > > cos sum formula itself, hinted by C-K_Shene:
> > > > Start with Euler's formula:
> > > > e^(i(x + y)) = cos(x + y) + i sin(x + y)
> > > > Apply the property e^(i(x + y)) = e^(ix) * e^(iy):
> > > > cos(x + y) + i sin(x + y) = (cos(x) + i sin(x)) * (cos(y) + i sin(y))
> > > > Expand the right side:
> > > > cos(x + y) + i sin(x + y) = cos(x)cos(y) + i cos(x)sin(y) + i sin(x)cos(y) + i^2 sin(x)sin(y)
> > > > Combine like terms and use i^2 = -1:
> > > > cos(x + y) + i sin(x + y) = (cos(x)cos(y) - sin(x)sin(y)) + i (cos(x)sin(y) + sin(x)cos(y))
> > > > Now, by comparing the real and imaginary parts on both sides, we get:
> > > > cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
> > > > sin(x + y) = cos(x)sin(y) + sin(x)cos(y)
> > > > https://chat.openai.com/share/7ac61120-0b20-4682-901b-e58f6eb66361
> > > >
> > > > I only used ChatGPT as secretary, didn't rely on some
> > > > ideas of ChatGPT. The proofs are mine, since ChatGPT
> > > > always needed several attempts, and I guided ChatGPT.
> > > >
> > > > Possibly the best use case for ChatGPT, use it as secretary.
> > > > Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:03:47 UTC+1:
> > > > > On Tuesday, November 28, 2023 at 12:57:38 AM UTC+1, Mild Shock wrote:
> > > > >
> > > > > > Thanks again ChatGPT.
> > > > > > https://chat.openai.com/share/c1b17dfd-8c4d-4633-9081-8bdc62b338f1
> > > > > I already told you that this "proof" is bogus (and why).
> > > > Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:12:55 UTC+1:
> > > > > On Tuesday, November 28, 2023 at 1:08:33 AM UTC+1, Mild Shock wrote:
> > > > >
> > > > > > Two 18 years old students just discovered a proof of
> > > > > > Pythagoras that mathematicians said was impossible.
> > > > > > https://9gag.com/gag/aEq4BeO
> > > > > Did they already publish it in a peer reviewed journal?

Or 9 days after April Fools day? Forgot how to count.

Mild Shock schrieb am Dienstag, 28. November 2023 um 01:28:20 UTC+1:
> The two girls made it into scientific american:
> https://www.scientificamerican.com/article/2-high-school-students-prove-pythagorean-theorem-heres-what-that-means/
>
> But this was 10 days after April Fools day?

FF was on the verge of starting a shitstorm, but
reason seems to have prevailed. No idea. But he still
owes us further explanations, as he dropped:

> > > Apply the angle sum/difference identity for cosine:
> > > cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
>
> Die Frage ist aber, wie man diese Formel beweist/beweisen hat.
> Habe eben mal im Internet danach gesucht. Die Seiten, die ich
> gefunden habe, haben dazu Pythagoras benutzt.

Is there a grain of truth in it? This proof below uses
Pythagoras in that a rotation is considered and length
of chords (chord, lat.) is equated.

Proof of Sum and Difference Identities
https://mymission.lamission.edu/userdata/sargsye2/docs/Math%20240/Proof%20of%20the%20difference%20formula%20for%20cosine.pdf

Where does this occur in the Euler formula? Proofs
of this kind, on the other hand, are without
Pythagoras, or are they?

prove the "sum and difference formulas" in trigonometry?
https://math.stackexchange.com/a/1342

How would Euclid do the proof, and how would that be
translated into today's trigonometry. What causes
everything to “fit together”?

Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:31:46 UTC+1:
> On Tuesday, November 28, 2023 at 1:28:20 AM UTC+1, Mild Shock wrote:
> > The two girls made it into scientific american:
> > https://www.scientificamerican.com/article/2-high-school-students-prove-pythagorean-theorem-heres-what-that-means/
> Yes. But I didn't see any details of their alleged proof.Maybe it isn't/wasn't that original. Who knows. :-P
>
> https://img-comment-fun.9cache.com/media/anzOg6o/aDXJ5A3J_700w_0.jpg
>
> No fake! [...]

The French Wikipedia has some excellent historic resources (Euclid etc..):
https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Pythagore

Mild Shock schrieb am Donnerstag, 30. November 2023 um 13:25:20 UTC+1:
> FF was on the verge of starting a shitstorm, but
> reason seems to have prevailed. No idea. But he still
> owes us further explanations, as he dropped:
> > > > Apply the angle sum/difference identity for cosine:
> > > > cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
> >
> > Die Frage ist aber, wie man diese Formel beweist/beweisen hat.
> > Habe eben mal im Internet danach gesucht. Die Seiten, die ich
> > gefunden habe, haben dazu Pythagoras benutzt.
>
> Is there a grain of truth in it? This proof below uses
> Pythagoras in that a rotation is considered and length
> of chords (chord, lat.) is equated.
>
> Proof of Sum and Difference Identities
> https://mymission.lamission.edu/userdata/sargsye2/docs/Math%20240/Proof%20of%20the%20difference%20formula%20for%20cosine.pdf
>
> Where does this occur in the Euler formula? Proofs
> of this kind, on the other hand, are without
> Pythagoras, or are they?
>
> prove the "sum and difference formulas" in trigonometry?
> https://math.stackexchange.com/a/1342
>
> How would Euclid do the proof, and how would that be
> translated into today's trigonometry. What causes
> everything to “fit together”?
> Fritz Feldhase schrieb am Dienstag, 28. November 2023 um 01:31:46 UTC+1:
> > On Tuesday, November 28, 2023 at 1:28:20 AM UTC+1, Mild Shock wrote:
> > > The two girls made it into scientific american:
> > > https://www.scientificamerican.com/article/2-high-school-students-prove-pythagorean-theorem-heres-what-that-means/
> > Yes. But I didn't see any details of their alleged proof.Maybe it isn't/wasn't that original. Who knows. :-P
> >
> > https://img-comment-fun.9cache.com/media/anzOg6o/aDXJ5A3J_700w_0.jpg
> >
> > No fake! [...]