Consider a regular polygon with $n$ sides, each of length $1$.

The distance between any two adjacent vertices is, of course, just $1$.

But what is the distance between any two "adjacent-but-one" vertices? (That is, two vertices that have exactly one other vertex between them.)

Is there any general way to find this? (I know that this is $2\cos \pi/n$ but I was hoping to express this distance solely in terms of $n$ and without using any trigonometric functions.)

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For example, in the case of the pentagon ($n=5$), we can find that it's $(1+\sqrt{5})/2$. So I was wondering if there are more general methods for finding the same for bigger $n$.