If a variable line in two adjacent positions has directions cosines $(l,m,n)(l+\delta l,m+\delta m,n+\delta n)$  and  $\delta \theta$ is the angle between the two positions then  ${\left(\delta l\right)}^2+{\left(\delta m\right)}^2+{\left(\delta n\right)}^2=$ 

$$\begin{gathered} l^2+m^2+n^2=1 --\left(1\right) \\ {(l+\delta l)}^2+{(m+\delta m)}^2+{(n+\delta n)}^2=1 \\ (l^2+m^2{n}^2)+[{\left(\delta l\right)}^2+{\left(\delta m\right)}^2+{\left(\delta n\right)}^2]+2[l\delta l+m\delta m+n\delta n]=1 \\ \Rightarrow {\left(\delta l\right)}^2+{\left(\delta m\right)}^2+{\left(\delta n\right)}^2=-2(l\delta l+m\delta m+n\delta n) \\ \therefore \cos \left(\delta \theta \right)=l(l+\delta l)+m(m+\delta m)+n(n+\delta n) \\ \cos \left(\delta \theta \right)=1+l\delta l+m\delta m+n\delta n \\ l\delta l+m\delta m+n\delta n=-2{\sin }^2\frac{\delta \theta }{2}=-2{\left(\frac{\delta \theta }{2}\right)}^2 \\ Approximately =-\frac{{\left(\delta \theta \right)}^2}{2} --------\left(4\right) \\ \left(3\right)\&\left(4\right) \Rightarrow {\left(\delta l\right)}^2+{\left(\delta m\right)}^2+{\left(\delta n\right)}^2=-2[-\frac{{\left(\delta \theta \right)}^2}{2}]={\left(\delta \theta \right)}^2 \end{gathered}$$