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?

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