\[ \hat{n}_1 + \hat{n}_2 \text{ are unit vectors} \] \[ \text{We have} \] \[ (|\hat{n}_1 + \hat{n}_2|)^2 = |\hat{n}_1|^2 + |\hat{n}_2|^2 + 2|\hat{n}_1||\hat{n}_2|\cos\theta \] \[ = 2(1 + \cos\theta) \] \[ \text{By rearranging } \cos\theta \text{ in terms of } \cos\frac{\theta}{2}, \text{ we get} \] \[ = 4\cos^2\frac{\theta}{2} \] \[ \therefore \cos\frac{\theta}{2} = \frac{1}{2}|\hat{n}_1 + \hat{n}_2| \]