I was asked to solve the following equation:

$\displaystyle 2sin(x) + csc(x) = 3$

I solved it by changing the csc(x) to $\displaystyle \frac{1}{sin(x)}$ and then by multiplying through by sin(x), yielding:

$\displaystyle 2sin^2(x) - 3sin(x) + 1 = 0$

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?

Thanks!