Sum to infinity - sector

Question

How would I start this question?

Answer 1

A Start: Note that the radius of the second circle is $\cos\theta$. So by similarity, if the first arclength is $a$, the second is $a\cos\theta$, the third is $a\cos^2\theta$, the fourth is $a\cos^3\theta$, and so on. The sum is an infinite geometric series.

Answer 2

Let $s_n$ denote the length of $A_n B_n$. We assume $A$ is $A_0$ and $B$ is $B_0$. We have $OA_n = \dfrac{s_n}{\theta}$. Hence, $OB_{n+1} = OA_n \cos(\theta)$. Hence, $$s_{n+1} = \theta \cdot OB_{n+1} = \theta \cdot OA_n \cos(\theta) = s_n \cos(\theta)$$ Hence, $s_n = \cos^n(\theta) s_0$, where $s_0 = 1$. Hence, $$A_0B_0 + A_1B_1 + \cdots = 1+ \cos(\theta) + \cos^2(\theta) + \cdots = \dfrac1{1-\cos(\theta)} = \dfrac{\csc^2(\theta/2)}2$$