let $\overrightarrow{a},\overrightarrow{b}\text{ and }\overrightarrow{c}$ be three unit vectors such that $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{\sqrt{3}}{2}(\overrightarrow{b}+\overrightarrow{c})$.if $\overrightarrow{b}$  is not parallel to $\overline{c}$, then angle between $\overrightarrow{a}\text{ and }\overrightarrow{c}$ is

given $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right|=1$
and $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{\sqrt{3}}{2}(\overrightarrow{b}+\overrightarrow{c})$
$\Rightarrow (\overrightarrow{a}.\overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{c}=\frac{\sqrt{3}}{2}\overrightarrow{b}+\frac{\sqrt{3}}{2}\overrightarrow{c}$
on comparing, we get
$\begin{array}{r}\overrightarrow{a}.\overrightarrow{c}=\frac{\sqrt{3}}{2}\Rightarrow |\overrightarrow{a}\parallel \overrightarrow{c}|\cos \theta =\frac{\sqrt{3}}{2}\\ \Rightarrow \cos \theta =\cos \frac{\pi }{6}\Rightarrow \theta =\frac{\pi }{6}\end{array}$