Answer
$-\dfrac {9\sqrt {2}}{8}$
Work Step by Step
$sec\left( 2\sin ^{-1}\left( -\dfrac {1}{3}\right) \right) =\dfrac {1}{\sin \left( 2\sin ^{-1}\left( -\dfrac {1}{3}\right) \right) }=\frac{1}{2\sin \left( \sin ^{-1}\left( -\dfrac {1}{3}\right) \right) \cos \left( \sin ^{-1}\left( -\dfrac {1}{3}\right) \right)} =\dfrac {1}{2\times \left( -\dfrac {1}{3}\right) \times \sqrt {1-\left( -\dfrac {1}{3}\right) ^{2}}}=\dfrac {1}{-\dfrac {4\sqrt {2}}{9}}=-\dfrac {9\sqrt {2}}{8}$