How to prove this trigonometric expression? How would you go about proving the following?
$${1- \cos A \over \sin A } + { \sin A \over 1- \cos A} = 2 \operatorname{cosec} A $$
This is what I've done so far:
$$LHS = {1+\cos^2 A -2\cos A + 1 - \cos^2A \over \sin A(1-\cos A)}$$
....no idea how to proceed .... X_X

You did everything thus far correctly, I just pick up with where you left off in the second line:
$$\begin{align}(1 - \cos A)^2 + \sin^2 A \over \sin A(1 - \cos A) 
& = \dfrac{1 - 2 \cos A + \cos^2 A + \sin^2 A}{\sin A(1 - \cos A)} \\ \\
& = {1 \color{blue}{\bf + \cos^2 A} -2\cos A + 1 \color{blue}{\bf - \cos^2A} \over \sin A(1-\cos A)} \\ \\
& = \dfrac{2 - 2\cos A}{\sin A(1 - \cos A)}\\ \\
& = \dfrac{2\color{red}{\bf (1-\cos A)}}{\sin A\color{red}{\bf (1 - \cos A)}}\\ \\
& = \frac{2}{\sin A} \\ \\
& = 2 \csc A
\end{align}$$
