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 04:32:14 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?

On Sunday, November 26, 2023 at 7:43:05 PM UTC-8, 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)
>
> 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 04:32:14 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?

Sounds like "identity dimension".

The identity-dimension, is a great sort of idea like this.

So, you know zero, is, the multiplicative annihilator,
and, that the singular points of the hypergeometric
are 0, 1, and infinity.

So, zero shows up a lot, being that
it's non-negative and non-positive,
it's right in the middle of things.

Then, identity-dimension,
is the idea for a diagram
on a Cartesian plane,
only employing the first quadrant
of positive (non-negative) numbers.

This is like f(x) = x, y = 1x + 0, x = y = z = ...,
there's the ray out from zero that makes for the
symmetry, among all its "co-dimensions".

So, this way it fits into analysis
as the envelope of the linear fractional
equation, Clairaut's, d'Alembert's, a bunch
of sorts of these integral equations whose
plane curves are lines, for example.

Lots of functions are symmetrical about it,
like 1/x the hyperbola's, and x^2+y^2=1 the circle.

So, it gets about as important as zero up in there.

About trigonometry, is tri-lateralometry,
it's starts with an equilateral triangle,
its base on a line, left-to-right, say.
Unhinge it on the left, then evolve each
of the angles open until it's a line.
Now, flip it forward and roll it back up.
The loose end just traced the sinusoid.