Your intermediate step is fine. But the only way a fraction can be equal to 0 is if the numerator is 0.
I have made attempts to find the solutions to th eeqution 2 sin x + csc x = 0 on the interval [0, 2pi). So, in my disconcertion, I flipped to the answer key and found the steps to finding solution. The steps are as follows:
2 sin x + csc X = 0 ---> 2 sin x + 1/sin x = 0
2 sin x + 1 = 0
What did they do to procure step three? Where did the denominator go? On that step I had gotten (2 sin x + 1)/sin x.