Which of the following relations is true for two unit vector A ^ and B ^ making an angle θ to each other?

We know,

 $$\begin{gathered} 1- \cos \theta =2{\sin }^2\left(\frac{\theta }{2}\right) \\ 1+ \cos \theta =2{\cos }^2\left(\frac{\theta }{2}\right) \end{gathered}$$

Now,$$\begin{gathered} \left(\overrightarrow{a}+\overrightarrow{b}\right)·\left(\overrightarrow{a}+\overrightarrow{b}\right)={\left|\overrightarrow{a}\right|}^2+2\overrightarrow{a}·\overrightarrow{b}+{\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow \left(\overrightarrow{a}+\overrightarrow{b}\right)·\left(\overrightarrow{a}+\overrightarrow{b}\right)={\left(1\right)}^2+2\times 1\times 1\times \cos \theta +{\left(1\right)}^2 \\ \Rightarrow \left(\overrightarrow{a}+\overrightarrow{b}\right)·\left(\overrightarrow{a}+\overrightarrow{b}\right)=2\left(1+\cos \theta \right) \\ \Rightarrow {\left|\overrightarrow{a}+\overrightarrow{b}\right|}^2=4{\cos }^2\left(\frac{\theta }{2}\right) \\ \Rightarrow \left|\overrightarrow{a}+\overrightarrow{b}\right|=2\cos \left(\frac{\theta }{2}\right) ...\left(1\right) \end{gathered}$$

Similarly,

$$\begin{gathered} \left(\overrightarrow{a}-\overrightarrow{b}\right)·\left(\overrightarrow{a}-\overrightarrow{b}\right)={\left|\overrightarrow{a}\right|}^2-2\overrightarrow{a}·\overrightarrow{b}+{\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow \left(\overrightarrow{a}-\overrightarrow{b}\right)·\left(\overrightarrow{a}-\overrightarrow{b}\right)={\left(1\right)}^2-2\times 1\times 1\times \cos \theta +{\left(1\right)}^2 \\ \Rightarrow \left(\overrightarrow{a}-\overrightarrow{b}\right)·\left(\overrightarrow{a}-\overrightarrow{b}\right)=2\left(1-\cos \theta \right) \\ \Rightarrow {\left|\overrightarrow{a}-\overrightarrow{b}\right|}^2=4{\sin }^2\left(\frac{\theta }{2}\right) \\ \Rightarrow \left|\overrightarrow{a}-\overrightarrow{b}\right|=2\sin \left(\frac{\theta }{2}\right) ...\left(2\right) \end{gathered}$$

Therefore,

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