#### 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.
$$\cos\frac{\theta}{2}=\frac{OH}{OC}$$
$$\cos\frac{\theta}{2}=\frac{R-b}{R}$$