Answer
$$\frac{5}{{2\sqrt 6 }}$$
Work Step by Step
$$\eqalign{
& \sec \left( {{{\sin }^{ - 1}}\left( { - \frac{1}{5}} \right)} \right) \cr
& = \sec \left( {{{\sin }^{ - 1}}\left( {\frac{1}{5}} \right)} \right) \cr
& {\text{Let }}\theta = {\sin ^{ - 1}}\left( {\frac{1}{5}} \right){\text{, thus}} \cr
& \sin \theta = \frac{1}{5} \cr
& {\text{Recall that }}\sin \theta = \frac{{{\text{opposite side}}}}{{{\text{hypotenuse}}}} \cr
& {\text{opposite side}} = 1 \cr
& {\text{hypotenuse}} = 5 \cr
& {\text{adjacent side}} = \sqrt {25 - 1} = 2\sqrt 6 \cr
& {\text{Then}} \cr
& \sec \theta = \sec \left( {{{\sin }^{ - 1}}\left( {\frac{1}{5}} \right)} \right) = \frac{{{\text{hypotenuse}}}}{{{\text{adjacent side}}}} \cr
& \sec \left( {{{\sin }^{ - 1}}\left( { - \frac{1}{5}} \right)} \right) = \frac{5}{{2\sqrt 6 }} \cr} $$