Let $\hat{a} $ and $\hat{b}$ are two unit vectors.
Then, $|\hat{a}+\hat{b}| =1 $
$ \Rightarrow |\hat{a}+\hat{b}|^{2} =1 $
$ \Rightarrow (\hat{a}+\hat{b}) \cdot(\hat{a}+\hat{b}) =1 $
$\Rightarrow |\hat{a}|^{2}+|\hat{b}|^{2}+2 \hat{a} \cdot \hat{b} =1 $
$\Rightarrow 1+1+2 \hat{a} \cdot \hat{ b } =1 $
$ \Rightarrow 2 \hat{a} \cdot \hat{b} =-1$
Again, $|\hat{ a }-\hat{ b }|^{2}=(\hat{ a }-\hat{ b }) \cdot(\hat{ a }-\hat{ b })$
$=|\hat{ a }|^{2}+| b |^{2}-2 \hat{ a } \cdot \hat{ b }$
$=1+1-(-1)$
$=1+1+1=3$
$\therefore |\hat{ a }-\hat{ b }|=\sqrt{3}$