If the sum of two unit vectors is a unit vector, then the magnitude of their difference is

$\sqrt{3}$

$|\hat{a}+\hat{b}|=1$

so  $(\hat{a}+\hat{b}) . (\hat{a}+\hat{b})=1 \Rightarrow 1+2 \hat{a} . \hat{b}+1=1$

so  $\hat{a} . \hat{b} = \frac{-1}{2}$

so  $\sqrt{(\hat{a}-\hat{b})(\hat{a}-\hat{b})}=\left|\hat{a}-\hat{b}\right| $

$=\sqrt{1-2 \hat{a} \hat{b}+1}=\sqrt{1+1-2 \times(\frac{-1}{2})}=\sqrt{3}$

Option: C