Thread by @TamasGorbe

Maclaurin series of trigonometric functions
– 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑝𝑟𝑜𝑜𝑓 –

The terms in the Maclaurin series

cos(x) = 1 – x²/2! + x⁴/4! – ...
sin(x) = x – x³/3! + x⁵/5! – ...

are the (signed) lengths of involutes. Here is the sketch of an elementary proof.

Here is the link to the video on youtube

I call the proof elementary, because it only uses

basic trigonometry: the base of an isosceles triangle with apex angle θ and legs of length L has length 2L×sin(θ/2)

a bit of combinatorics: binomial coeff's

and a tiny piece of calculus: sin(t)/t converges to 1 as t goes to 0

The earliest reference to this proof I could find is by Leo Gurin who attributes it to his teacher Yakov Chaikovsky and dates it to 1935.

I would be amazed if that was the first time someone came up with this proof.

For more details, read Gurin's paper
jstor.org/stable/2974881
or this answer on Math Stack Exchange:
math.stackexchange.com/a/2758743