Show that $\;$ $\sqrt{\dfrac{1 - \cos A}{1 + \cos A}} = \dfrac{1 - \cos A}{\sin A}$
$\begin{aligned}
LHS & = \sqrt{\dfrac{1 - \cos A}{1 + \cos A}} \\\\
& = \sqrt{\dfrac{\left(1 - \cos A\right)^2}{\left(1 + \cos A\right) \left(1 - \cos A\right)}} \\\\
& = \dfrac{1 - \cos A}{\sqrt{1 - \cos^2 A}} \\\\
& = \dfrac{1 - \cos A}{\sqrt{\sin^2 A}} \\\\
& = \dfrac{1 - \cos A}{\sin A} = RHS
\end{aligned}$
Hence proved.