Removing squared values by dividing I am solving a simple trig equation:
$$
2\cos^2{\theta} + \cos{\theta} = 0
$$
My first approach was to move the $\cos\theta$ to the other side, and then divide by $\cos\theta$.
$$
2\cos^2{\theta} = -\cos{\theta}
$$
$$
2\cos{\theta} = -1
$$
$$
\cos{\theta} = -\frac{1}{2}
$$
$$
\theta = \frac{2\pi}{3}, \frac{4\pi}{3}
$$
But apparently this approach causes me to miss some of the solution. The correct approach seems to be to extract $\cos\theta$ and solve like a quadratic:
$$
\cos{\theta}(2\cos{\theta}+1) = 0
$$
$$
\cos{\theta} = 0, -\frac{1}{2}
$$
$$
\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}
$$
I can kind of see that since $\cos\theta$ is squared, $\cos\theta$ could be $\pm$, but I'm not sure what it means for dividing by $\cos\theta$. Is it possible to come to the correct solution by dividing, or do I need to steer clear of that when it comes to squared values?

The point is only that if $\cos\theta=0$ then you cannot divide both sides by $\cos\theta;$ i.e. you cannot divide both sides by $0.$ Note, for example that it is true that $3\times0=5\times0,$ but dividing both sides by $0$ and getting $3=5$ is wrong since $3\ne5.$
So dividing both sides by $\cos\theta$ gives you all solutions for which $\cos\theta\ne0,$ but then you also have to check whether there are any solutions for which $\cos\theta=0.$
If $AB=0,$ then either $A=0$ or $B=0.$ You have $A=\cos\theta$ and $B=2\cos\theta+1.$
