When solving $\sin(2\theta) = \cos\theta$ for $\theta$ for all values in the range $[0,2\pi]$ I only get half of the solutions when I reduce $\sin(2\theta) = \cos\theta$ to $\sin\theta = \frac{1}{2}$.
So before reducing the equation the solutions are $\frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{3\pi}{2}$ and after reducing the equation the solutions are only $\frac{\pi}{6}, \frac{5\pi}{6}$
I can see how it reduces the solution set, but I want to know 'why' it happens.
How can you know exactly when you're reducing the solution set of an equation when you're reducing the equation itself to a simpler form, especially when it's a big complicated equation when you can't easily see that you're reducing the solution set as in this case?
Is there some information regarding that or am I asking the wrong question?