E686 -- Dilucidationes super formulis, quibus sinus et cosinus angulorum multiplorum exprimi solent, ubi simul ingentes difficultates diluuntur

(Elucidations about the formula, in which the sines and cosines of angles are to be multiplied, where at once large difficulties are diluted)


Euler starts with a polynomial expansion of 2 cos(nf). He then substitutes for z = cos f and s = cos(nf) and gets a second order differential equation. He tries to solve that with unknown coefficients and gets another series for cos(nf).

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