>After searching this site (to no avail), does there exist a
>way to show "the derivative of the sine is the cosine" that
>doesn't rely on already knowing the derivative of the sine?
>Using the Taylor expansion it's easy, but that relies on the
>derivative... There is a little something that at
https://www.cut-the-knot.org/Curriculum/Calculus/SineCosine.shtml
that my suggest an alternative proof. A rather standard proof by definition of the derivative uses only
- The sine of sum formula (https://www.cut-the-knot.org/proofs/sine_cosine.shtml)
- The continuity of sine and cosine
- The limit of sin(x)/x.
>
>Also, I'm familiar with the density theorems for the real
>numbers (between any two distinct rational/irrational
>numbers there exists an irrational/rational number), but can
>one say that adjacent to any rational/irrational number are
>two irrational/rational numbers? Or does the notion of
>adjacency not exist on the real line?
No, it does not.
https://www.cut-the-knot.org/do_you_know/numbers.shtml
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