# How do you prove  (cosA - cotA) / ( 1 - sinA) = - cotA?

Use that $\cot A = \frac{\cos A}{\sin A}$ in the left hand side and then factor the expression.
$\frac{\cos A - \cot A}{1 - \sin A} = \frac{\cos A - \frac{\cos A}{\sin A}}{1 - \sin A} = \cos A \left(\frac{1 - \frac{1}{\sin A}}{1 - \sin A}\right)$
$= \frac{\cos A}{\sin A} \left(\frac{\sin A - 1}{1 - \sin A}\right) = \frac{\cos A}{\sin A} \cdot \left(- 1\right) = - \cot A$