Why dividing by trigonometric functions gives wrong answer when solving trigonometric equations?

Hello I have a problem with solving Trigonometric equations. Why this is not true for $0\le\theta\le360$

$$2\sin\theta\cos\theta=\sin\theta$$
$$2\cos\theta=1$$
Set of solutions $\theta=60,360$

and this is ?

$$2\sin\theta\cos\theta=\sin\theta$$
$$\sin\theta(2\cos\theta-1)=0$$
Set of solutions $\theta=0,60,180,300,360$

Both of those seem to be logical and yet they give different results. Can you tell me when should I notice that I can’t divide by $\sin\theta$ or some other trig?

Solutions Collecting From Web of "Why dividing by trigonometric functions gives wrong answer when solving trigonometric equations?"

$\sin \theta$ can be $0$. Thus, you divide by zero.

If $\sin{\theta}=0$, your equation is satisfied whatever value $2\cos{\theta}$ has (not necessarily $1$), which gives you some values of $\theta$. So you have to exclude the solutions to $\sin{\theta}=0$ before you divide by it, which will give you other values.

Whenever you want to cancel a factor, check whether it might be zero.