Answer
$$\cos\frac{\theta}{2}=\frac{R-b}{R}$$
Work Step by Step
(The image is shown below)
We see in the image that $BC$ is a circular curve where $OB=OC=R$. That means $OA=R$ as $A$ is a point in the circular curve.
Thus, $OH=OA-AH=R-b$
As triangle $OHC$ is a right triangle, we can calculate $\cos\frac{\theta}{2}$ using the sides of the triangle.
$$\cos\frac{\theta}{2}=\frac{OH}{OC}$$
$$\cos\frac{\theta}{2}=\frac{R-b}{R}$$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/45a60538-d855-487b-88e0-776db04b0d5c/steps_image/small_1498080623.png?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T012501Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=c7fee4e12b221b4aa8119d54848380cc61e6787925f026cfac0da238f2a63308)