sin²(x)+cos²(x) = 1 isn't 𝑗𝑢𝑠𝑡 Pythagorean theorem on the unit circle It also tells us that a rotation (eⁱˣ) followed by its inverse rotation (e⁻ⁱˣ) is the same as no rotation at allpic.twitter.com/INSZcTxtXr
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And HOW do you know this without knowing that the equation of the circle is x^2+y^2=1?
I don’t get your point. All you need is the fact that they are equivalent, which you get from the ODE.
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When you compute the derivative of sin and cos (of the geometric functions, I mean) you are using the properties of the circle in order to approximate the arcs, hence (implicitly) Pithagorean theorem.
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You need to know that sin(t)/t tends to 1 when t goes to 0. Otherwise you cannot differentiate sin. And you *need* trigonometry in order to know this. And trigonometry IS Pythagorean identity.
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Writing down your ODE for the geometric circular functions requires Pythagorean theorem at the very beginning.
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