The following inequality, ascribed to Huygens, appeared in this post:

\begin{equation*}
1-\frac43\,\frac{\sin^3\theta/2}{\theta-\sin\theta}
>(1-\cos\theta/2)\Big(\frac35-\frac3{1400}\frac{\pi^2}{n^2}\Big), \tag{10}\label{10}
\end{equation*}
where $\theta\in[0,\pi/2]$ and $n$ is an integer $\ge2$.
Later, this inequality was changed to its opposite.
Note that neither the strict inequality \eqref{10} nor its strict opposite hold for $\theta=0$.
It was also said that "the inequality as stated here is the beginning of the asymptotic expansion of the quotient of the left-hand side by the trigonometric function on the right".

The mysterious role of $n$ here has not been explained. However, one has \begin{equation*} f(\theta):=\Big(1-\frac43\,\frac{\sin^3\theta/2}{\theta-\sin\theta}\Big) \Big/(1-\cos\theta/2) =\frac35-\frac{3\theta^2}{1400}-\frac{11\theta^4}{336000}+O(\theta^6). \end{equation*} So, it makes sense to ask if \begin{equation*} f(\theta)<\frac35-\frac{3\theta^2}{1400} \quad\text{for }\theta\in(0,\pi/2]. \tag{20}\label{20} \end{equation*}

It will be shown below that this is indeed true.