<s>[INST] Verify this identity: $\sin x/(1 - \cos x) = \csc x + \cot x$ Verify this identity $\frac{\sin {x}} {1 - \cos {x}}\ =  \csc x + \cot x$  I got the left side to $\frac{1-\cos x} {\csc x}$, but I can't get any farther.  Am I on the right path?  Can I get some help?
 [/INST] The right-hand side can be rewritten as
$$\csc{x} + \cot{x} = \frac{1}{\sin{x}} + \frac{\cos{x}}{\sin{x}} = \frac{1 + \cos{x}}{\sin{x}}$$
Multiplying the left side top and bottom by $1 + \cos{x}$, we find that
$$\frac{\sin{x}}{1 - \cos{x}} = \frac{\sin{x}(1 + \cos{x})}{1 - \cos^2{x}} = \frac{\sin{x}(1 + \cos{x})}{\sin^2{x}} = \frac{1 + \cos{x}}{\sin{x}}$$
</s>