Elementary derivation of the Taylor series of $\sin x$ that might have been devised by Madhava of Sangamagrama

Question

I can derive the Taylor series expansion of $\sin x$ by the Taylor's theorem.

But what is an elementary approach to derive this series? I mean the way Madhava of Sangamagrama (c.1340–c.1425) or his contemporaries had possibly derived.

Answer 1

I suppose, Madhava of Sangamagrama had enough of calculus available to know what we would write as $\frac {\mathrm d}{\mathrm dx}\sin x=\cos x$ and $\frac {\mathrm d}{\mathrm dx}\cos x=\sin x$, hence $\frac {\mathrm d^2}{\mathrm dx^2}\sin x=-\sin x$ as well as $\frac {\mathrm d}{\mathrm dx}x^n=nx^{n-1}$. Then comparing coefficients of a series ansatz $$\sin x=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+\ldots $$ with its second derivative together with the elementary observation that $a_0=0$ and $a_1=1$ (i.e., $\sin x\approx x$ for small $x$) directly leads to $$\sin x=x-\frac16x^3+\frac1{120}x^5\mp\ldots $$ though the question of convergence may not have been dealt with rigorously (beyond justification by numerical evidence perhaps).