Answer
$-2sec^2\alpha $
Work Step by Step
$\dfrac {1}{\sin \alpha -1}-\dfrac {1}{\sin \alpha +1}=\dfrac {\left( \sin \alpha +1\right) -\left( \sin \alpha -1\right) }{\left( \sin \alpha -1\right) \left( \sin \alpha +1\right) }=\dfrac {2}{-\left( 1-\sin ^{2}\alpha \right) }=\dfrac {2}{-\cos ^{2}\alpha }=-2sec^2\alpha
$