There isn't a simpler method of solution. However, there is another solution to your quadratic. (The solution you found is one solution.)
I was asked to solve the following equation:
I solved it by changing the csc(x) to and then by multiplying through by sin(x), yielding:
I then used the "completing the square" method to solve the equation and arrived at sin(x) = 1. Because I ended up with such a neat answer, I'm wondering if I missed some easier way to solve the problem.
Was there a more simple way I could have solved this that wouldn't have involved completing the square or using the quadratic equation?