Answer
The identity is verified.
$\frac{1}{cos~x+1}+\frac{1}{cos~x-1}=-2~csc~x~cot~x$
Work Step by Step
$\frac{1}{cos~x+1}+\frac{1}{cos~x-1}=\frac{1}{cos~x+1}~\frac{cos~x-1}{cos~x-1}+\frac{1}{cos~x-1}~\frac{cos~x+1}{cos~x+1}=\frac{cos~x-1+cos~x+1}{cos^2x-1}=-\frac{2~cos~x}{1-cos^2x}=-\frac{2~cos~x}{sin^2x}=-2\frac{1}{sin~x}\frac{cos~x}{sin~x}=-2~csc~x~cot~x$
