What am I doing wrong? (Trigonometric Identity) $$\frac { \cos\theta  }{ 1-\sin\theta  } =\frac { \sin\theta -\csc\theta  }{ \cos\theta -\cot\theta  } $$
Steps I took:
$$\frac { \sin\theta -\frac { 1 }{ \sin\theta  }  }{ \cos\theta -\frac { \cos\theta  }{ \sin\theta  }  } $$
$$\frac { \frac { \sin^{ 2 }\theta -1 }{ \sin\theta  }  }{ \frac { \sin\theta \cos\theta -\cos\theta  }{\sin\theta  }  } $$
$$\frac { \sin^{ 2 }\theta -1 }{ \sin\theta  } \cdot \frac { \sin\theta  }{ \sin\theta \cos\theta -\cos\theta  } $$
$$\frac { (\sin^{ 2 }\theta -1) }{ (\cos\theta )(\sin\theta -1) } $$
Now where do I go from here? As far as I can tell, $\sin^{ 2 }\theta -1=-\cos^2\theta $ 
so this makes no sense to me.

You've already solved the problem:
$$\begin{align}
\dfrac { (\sin^{ 2 }\theta -1) }{ (\cos\theta )(\sin\theta -1) } &= \dfrac{-\cos^2\theta}{(\cos\theta)(\sin\theta - 1)} \\
&= \dfrac{-\cos\theta}{\sin\theta-1} \\
&= \dfrac{\cos \theta}{-(\sin\theta-1)} \\
&= L.H.S.
\end{align}$$
