The arc length AB in a circle of radius 1 with sector angle \( \theta \) is \( \theta \) A1.

Since (AB₁) is perpendicular to OB, the radius to B₁ is OB₁ = \( \cos \theta \). Thus, arc A₁B₁ at radius \( \cos \theta \) has length \( \theta \cos \theta \) A1.

Similarly, (A₁B₂) is perpendicular to OB, so the radius to B₂ is \( \cos \theta \cdot \cos \theta = \cos^2 \theta \). Arc A₂B₂ has length \( \theta \cos^2 \theta \). This continues, with arc A_nB_n having length \( \theta \cos^n \theta \).

The sum of the arc lengths is \( \theta + \theta \cos \theta + \theta \cos^2 \theta + \theta \cos^3 \theta + \dots = \theta \sum_{n=0}^\infty \cos^n \theta \) M1.

This is a geometric series with first term 1 and common ratio \( \cos \theta \). The sum to infinity is \( \sum_{n=0}^\infty \cos^n \theta = \frac{1}{1 – \cos \theta} \), so the total sum is \( \theta \cdot \frac{1}{1 – \cos \theta} = \frac{\theta}{1 – \cos \theta} \) A1.

[4 marks]