Let $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$   be three unit vectors such that $$\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = \frac{{\overrightarrow b + \overrightarrow c }}{{\sqrt 2 }}$$      and the angles between $$\overrightarrow a ,\,\overrightarrow c $$  and $$\overrightarrow a ,\,\overrightarrow b $$  be $$\alpha $$ and $$\beta $$ respectively then :

Solution :
$$\eqalign{ & \overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = \frac{{\overrightarrow b + \overrightarrow c }}{{\sqrt 2 }}\,\,\,\,\, \Rightarrow \,\left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow b - \left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow c = \frac{{\overrightarrow b + \overrightarrow c }}{{\sqrt 2 }}\, \cr & \Rightarrow \overrightarrow a .\overrightarrow c = \frac{1}{{\sqrt 2 }},\,\,\,\,\,\overrightarrow a .\overrightarrow b = - \frac{1}{{\sqrt 2 }} \cr & \Rightarrow \left| {\overrightarrow a } \right|\left| {\overrightarrow c } \right|\cos \,\alpha = \frac{1}{{\sqrt 2 }},\,\,\,\,\,\left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\cos \,\beta = - \frac{1}{{\sqrt 2 }} \cr & \therefore \,\,\cos \,\alpha = \frac{1}{{\sqrt 2 }},\,\,\,\,\,\cos \,\beta = - \frac{1}{{\sqrt 2 }} \cr & \therefore \alpha = \frac{\pi }{4},\,\,\,\,\,\beta = \frac{{3\pi }}{4} \cr} $$