Which of the following relations is true for two unit vectors $$\hat{A}$$ and $$\hat{B}$$ making an angle $$\theta$$ to each other?

$$|\hat{A} + \hat{B}| = \sqrt{|\hat{A}|^2 + |\hat{B}|^2 + 2|\hat{A}||\hat{B}|\cos\theta} = \sqrt{1 + 1 + 2\cos\theta} = \sqrt{2(1 + \cos\theta)}$$

$$\implies |\hat{A} + \hat{B}| = \sqrt{2 \cdot 2\cos^2\frac{\theta}{2}} = 2\cos\frac{\theta}{2} \quad \text{--- (1)}$$

$$|\hat{A} - \hat{B}| = \sqrt{|\hat{A}|^2 + |\hat{B}|^2 - 2|\hat{A}||\hat{B}|\cos\theta} = \sqrt{1 + 1 - 2\cos\theta} = \sqrt{2(1 - \cos\theta)}$$

$$\implies |\hat{A} - \hat{B}| = \sqrt{2 \cdot 2\sin^2\frac{\theta}{2}} = 2\sin\frac{\theta}{2} \quad \text{--- (2)}$$

$$\frac{|\hat{A} - \hat{B}|}{|\hat{A} + \hat{B}|} = \frac{2\sin\frac{\theta}{2}}{2\cos\frac{\theta}{2}} = \tan\frac{\theta}{2}$$

$$\implies |\hat{A} - \hat{B}| = |\hat{A} + \hat{B}|\tan\frac{\theta}{2}$$