$n$ small circles are tangent to each other and tangent to the big circle. Here's a figure for $n=4$:
Asking hints of how to find the reason between the radius of small circles into the big circle.
I tried doing some trigonometrics on the arcs, but failed.
Also tried this but I'm not sure about it:
Assuming $r$ is the small radius, $R$ is the radius of big one, $n$ is the amount of small circles inside the big and $x$ is the distance from the center of big circle to the center of small circle.
$$\sin\frac\pi n = \frac r x \tag{1}$$ But we don't know what $x$ is yet. We can get it from $$x + r = R \tag{2}$$ and combine the two equations: $$x = R - r = \frac{r}{\sin\frac\pi n} \tag{3}$$ $$R = r \left(1 + \frac{1}{\sin\frac\pi n} \right) \tag{4}$$
Therefore,
$$R \sin\frac\pi n = r \left(\sin\frac\pi n + 1 \right) \tag{5}$$
The centers of the small circles form a regular polygon with sides $2r$. You can find the circumradius of this polygon, then increase it by $r$ to find $R$. The circumradius is $\frac{2r}{\sin \frac \pi n}$ Then $R=\frac{2r}{2\sin \frac \pi n}+r$ which is equivalent to your $R\sin \frac \pi n=r(\sin \frac \pi n+1)$