Answer
$$\frac{7}{8}$$
Work Step by Step
$$\eqalign{
& \cos \left( {2{{\sin }^{ - 1}}\frac{1}{4}} \right) \cr
& {\text{Let }}\theta = {\sin ^{ - 1}}\frac{1}{4}{\text{, thus}} \cr
& \sin \theta = \frac{1}{4} \cr
& {\text{Recall that }}\sin \theta = \frac{{{\text{opposite side}}}}{{{\text{hypotenuse}}}} \cr
& {\text{opposite side }} = 1 \cr
& {\text{hypotenuse}} = 4 \cr
& {\text{adjacent side}} = \sqrt {15} \cr
& {\text{We have that }}\cos \left( {2{{\sin }^{ - 1}}\frac{1}{4}} \right) = \cos \left( {2\theta } \right) \cr
& {\text{Use the identity }}\cos 2\theta = 1 - 2{\sin ^2}\theta \cr
& \cos \left( {2{{\sin }^{ - 1}}\frac{1}{4}} \right) = 1 - 2{\left( {\frac{1}{4}} \right)^2} \cr
& \cos \left( {2{{\sin }^{ - 1}}\frac{1}{4}} \right) = \frac{7}{8} \cr} $$