Sin x divided by 1 minus cos x equals csc plus cot x?

It helps to convert this kind of equation into one that has only sines and cosines, by using the basic definitions of the other functions in terms of sines and cosines.

sin x / (1 - cos x) = csc x + cot x

sin x / (1 - cos x) = 1 / sin x + cos x / sin x

Now it should be easy to do some simplifications:

sin x / (1 - cos x) = (1 + cos x) / sin x

Multiply both sides by 1 + cos x:

sin x (1 + cos) / ((1 - cos x)(1 + cos x)) = (1 + cos x)2 / sin x

sin x (1 + cos) / (1 - cos2x) = (1 + cos x)2 / sin x

sin x (1 + cos) / sin2x = (1 + cos x)2 / sin x

sin x (1 + cos x) / sin x = (1 + cos x)2

1 + cos x = (1 + cos x)2

1 = 1 + cos x

cos x = 0

So, cos x can be pi/2, 3 pi / 2, etc.

In some of the simplifications, I divided by a factor that might be equal to zero; this has to be considered separately. For example, what if sin x = 0? Check whether this is a solution to the original equation.