Can you solve $\sin x=\sin(x/2)$ without a sum/double/half angle identity?

If you have the equation $$\sin x=\sin\frac{x}{2}$$ and you need to solve for $$x$$, such that $$0\leq x\leq 2\pi$$, can you do this without the use of any identities?

Someone asked me this question and I said that the way to solve could be by using the sum identity. First I subtracted $$\sin(\frac{x}{2})$$ from both sides and then applied the identity: $$-\sin(s)+\sin(t)=-2\cos(\frac{s+t}{2})\sin(\frac{s-t}{2})$$

So I simplified my equation to get $$2\cos(\frac{3x}{4})\sin(\frac{x}{4})=0$$

Then solving $$\cos(\frac{3x}{4})=0$$ and $$\sin(\frac{x}{4})=0$$, I get that $$x=\frac{2\pi}{3}, 2\pi, 0$$

However the person is not familiar with any identities and I wanted to know if there is a way to solve this without them.

• Note that if sin A = sin B, A=B + 2pik or A = 180 - B + 2pik – Gabe Nov 25 '19 at 22:12

$$\sin(x)=\sin\left(\frac{x}{2}\right) \quad \iff\quad x=\frac x2+2k\pi \quad \lor \quad x=\pi-\frac x2+2k\pi$$