<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] To prove sin x/1-cos x = cosec x + cot x
We know that,
Sin^2 x = 1- cos^2
So from RHS we get,
Cosec x+cot x
=1/sin x + cos x/sin x
So taking LCM,
We get,
1+cos x/sin x
Now multiply both numerator and denominator by sin x to get denominator as sin^2 x
We get,
(1+cos x)(sin x)/sin^2 x
Now we can write sin^2 x as 1- cos^2 x
So, we get 
(1+cos x)(sin x)/1-cos^2 x
(1+cos x)(sin x)/(1+cos x)(1-cos x)
applying a^2-b^2=(a+b)(a-b)
so by cancellation
We get sin x/1-cosx=cosec x+cot x
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