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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.

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    $\begingroup$ Note that if sin A = sin B, A=B + 2pik or A = 180 - B + 2pik $\endgroup$ – Gabe Nov 25 '19 at 22:12
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We can simply use that by definition

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

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