Step-by-step explanation:
[tex] \frac{ \sin(A) }{1 - \cos(A) } = \frac{ \sin(A)(1 + \cos(A) ) }{( {1}^{2} + { \cos }^{2}A) } \\ = \frac{\sin(A) + \sin(A) \cos(A) }{1 - \cos ^{2} (A) } \\ = \frac{\sin(A) + \sin(A) \cos(A) }{ \sin ^{2} (A) } \\ = \frac{1 + \cos(A) }{ \sin(A) } \\ for \: half \: angles \\ = \frac{1 + (2 \cos ^{2} ( \frac{A}{2} ) - 1)}{2 \sin( \frac{A}{2}) \cos( \frac{A}{2} ) } \\ = \frac{ { \cos }^{2} \frac{A}{2} }{ \sin( \frac{A}{2} ) \cos( \frac{A}{2} ) } \\ = \frac{ \cos( \frac{A}{2} ) }{ \sin( \frac{A}{2} ) } \\ = \cot( \frac{A}{2} ) [/tex]
