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