Write sin 2x Out

$\sin\left(2x\right) = \sin\left(x\right)$ mixes two angles, 2x and x. Write the double angle in terms of x with $\sin\left(2x\right) = 2\sin\left(x\right)\cos\left(x\right)$, and bring everything to one side: $2\sin\left(x\right)\cos\left(x\right)-\sin\left(x\right) = 0$.

Now sin x is a common factor: $\sin\left(x\right)\left(2\cos\left(x\right)-1\right) = 0$. So sin x = 0, giving 0° and 180°, or cos x = 1/2, giving 60° and 300°. Dividing by sin x at the start would have lost 0° and 180°.