Answer
$$\frac{4}{{\sqrt 7 }}$$
Work Step by Step
$$\eqalign{
& \sec \left[ {{{\sin }^{ - 1}}\left( { - \frac{3}{4}} \right)} \right] \cr
& {\text{Use si}}{{\text{n}}^{ - 1}}\left( { - \theta } \right) = - \sin \theta \cr
& = \sec \left[ { - {{\sin }^{ - 1}}\left( {\frac{3}{4}} \right)} \right] \cr
& {\text{Use }}\sec \left( { - \theta } \right) = \sec \theta \cr
& = \sec \left[ {{{\sin }^{ - 1}}\left( {\frac{3}{4}} \right)} \right] \cr
& {\text{From the triangle shown below }} \cr
& \sin \theta = \frac{3}{4} \cr
& \theta = {\sin ^{ - 1}}\frac{3}{4} \cr
& {\text{Therefore}} \cr
& \sec \left[ {{{\sin }^{ - 1}}\left( {\frac{3}{4}} \right)} \right] = \sec \theta \cr
& {\text{From the triangle}} \cr
& \sec \theta = \frac{4}{{\sqrt 7 }} \cr} $$
