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

$\overrightarrow{a}-\overrightarrow{b}=\overrightarrow{c }\left|\overrightarrow{a}+\overrightarrow{b}\right|=$

Let $\theta$ be the angle between $\widehat{a}and\widehat{b}$

$\widehat{a}-\widehat{b}=\widehat{c}$

1 = 1 + 1 + 2$\cos \left(180-\theta \right)$

0=1-2 $\cos \theta \Rightarrow \theta ={60}^0$

now $\left(\widehat{a}+\widehat{b}\right)=\sqrt{1+1+2\cos 60}=\sqrt{3}$