If $$\sin \left( {\theta + {{30}^ \circ }} \right) = \frac{3}{{\sqrt {12} }}{\text{,}}$$     then find $${\text{co}}{{\text{s}}^2}\theta ?$$

$$\eqalign{ & \sin \left( {\theta + {{30}^ \circ }} \right) = \frac{3}{{\sqrt {12} }} = \frac{3}{{2\sqrt 3 }} \cr & = \frac{{\sqrt 3 }}{2} = \sin \left( {\theta + {{30}^ \circ }} \right) = \sin {60^ \circ } \cr & \therefore \theta = {30^ \circ } \cr & {\text{co}}{{\text{s}}^2}\theta = {\text{co}}{{\text{s}}^2}{30^ \circ } \cr & = {\left( {\frac{{\sqrt 3 }}{2}} \right)^2} \cr & = \frac{3}{4} \cr} $$