Answer
$\frac{1}{csc~θ+1}-\frac{1}{csc~θ-1}=2~tan^2~θ$
Work Step by Step
$cot^2θ+1=csc^2θ$
$csc^2 θ-1=cot^2θ$
$\frac{1}{cot^2 θ}=tan~θ$
$\frac{1}{csc~θ+1}-\frac{1}{csc~θ-1}=\frac{1}{csc~θ+1}\frac{csc~θ-1}{csc~θ-1}-\frac{1}{csc~θ-1}\frac{csc~θ+1}{csc~θ+1}=\frac{csc~θ-1}{csc^2~θ-1}-\frac{csc~θ+1}{csc^2~θ-1}=\frac{-2}{csc^2~θ-1}=-\frac{2}{cot^2θ}=2~tan^2~θ$
