The longer way of writing it out #1

The given sum is $f(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...=\sin x\implies f'(x) =\cos x$

$\implies \frac{f'(x) }{f(x)}=\cot x$ .

$\implies \dfrac{f'(\frac{π}{6})}{f(\frac{π}{6})}=\cot (\frac{π}{6})=\sqrt 3\approx \boxed {1.732}$ .

$\begin{aligned} f(x) & = - \frac 1x \sum_\blue{n=1}^\infty \frac {(-1)^n x^{2n}}{(2n-1)!} \\ & = \frac 1x \sum_\red{n=0}^\infty \frac {(-1)^n x^{2n+2}}{(2n+1)!} \\ & = \sum_{n=0}^\infty \frac {(-1)^n x^{2n+1}}{(2n+1)!} \\ & = \sin x \\ \implies f'(x) & = \cos x \\ \frac {f'(\frac \pi 6)}{f(\frac \pi 6)} & = \cot \frac \pi 6 = \sqrt 3 \approx \boxed{1.73} \end{aligned}$