Answer
$$\frac{{\sqrt 2 + \sqrt 6 }}{4}$$
Work Step by Step
$$\eqalign{
& \cos \left( { - {{15}^ \circ }} \right) \cr
& {\text{Write 1}}{{\text{5}}^ \circ }{\text{ as }}{45^ \circ } - {60^ \circ } \cr
& = \cos \left( {{{45}^ \circ } - {{60}^ \circ }} \right) \cr
& {\text{Use cosine difference identity}} \cr
& = \cos \left( {{{45}^ \circ }} \right)\cos \left( {{{60}^ \circ }} \right) + \sin \left( {{{45}^ \circ }} \right)\sin \left( {{{60}^ \circ }} \right) \cr
& {\text{Substitute known values}} \cr
& = \left( {\frac{{\sqrt 2 }}{2}} \right)\left( {\frac{1}{2}} \right) + \left( {\frac{{\sqrt 2 }}{2}} \right)\left( {\frac{{\sqrt 3 }}{2}} \right) \cr
& {\text{Simplifying}} \cr
& = \frac{{\sqrt 2 }}{4} + \frac{{\sqrt 6 }}{4} \cr
& = \frac{{\sqrt 2 + \sqrt 6 }}{4} \cr} $$