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

$\left|\hat{\mathrm{A}} + \hat{\mathrm{B}}\right| = \sqrt{{\left|\hat{A}\right|}^2 + {\left|\hat{B}\right|}^2+ 2\left|\hat{A}\right|\left|\hat{B}\right| \cos \theta }$

$= \sqrt{1 + 1+ 2\mathrm{cos\theta }}$

$= \sqrt{2 (1+ \mathrm{cos\theta })}$

$=\sqrt{2 \mathrm{x} 2{\cos }^2\frac{\mathrm{\theta }}{2}}$

$=2\cos \frac{\mathrm{\theta }}{2}$

$\left|\hat{\mathrm{A}} - \hat{\mathrm{B}}\right| = \sqrt{{\left|\hat{A}\right|}^2 + {\left|\hat{B}\right|}^2- 2\left|\hat{A}\right|\left|\hat{B}\right| \cos \theta }$

$=\sqrt{2 - 2 \mathrm{cos\theta }}$

$=2\sin \frac{\mathrm{\theta }}{2}$

$=\frac{\left|\hat{\mathrm{A}} + \hat{\mathrm{B}}\right|}{\left|\hat{\mathrm{A}} - \hat{\mathrm{B}}\right|} = \cot \frac{\mathrm{\theta }}{2}$

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