Answer
The identity is verified.
$\frac{sec~\theta-1}{1-cos~\theta}$: red
$sec~\theta$: black
Work Step by Step
Remember:
$sec~\theta=\frac{1}{cos~\theta}$
So:
$sec~\theta~cos~\theta=\frac{1}{cos~\theta}~cos~\theta=1$
$\frac{sec~\theta-1}{1-cos~\theta}~\frac{1+cos~\theta}{1+cos~\theta}=\frac{sec~\theta+1-1-cos~\theta}{1-cos^2\theta}=\frac{sec~\theta-cos~\theta}{sin^2\theta}~\frac{cos~\theta}{cos~\theta}=\frac{1-cos^2\theta}{sin^2\theta~cos~\theta}=\frac{sin^2\theta}{sin^2\theta~cos~\theta}=\frac{1}{cos~\theta}=sec~\theta$
