[tex]\sin^2\dfrac x2=\sin^2x[/tex]
[tex]\dfrac{1-\cos x}2=1-\cos^2x[/tex]
[tex]1-\cos x=2-2\cos^2x[/tex]
[tex]2\cos^2x-\cos x-1=0[/tex]
[tex](2\cos x+1)(\cos x-1)=0[/tex]
[tex]\implies\begin{cases}2\cos x+1=0\\\cos x-1=0\end{cases}[/tex][tex]\implies\begin{cases}\cos x=-\frac12\\\cos x=1\end{cases}[/tex]
The first case occurs in [tex]0\le x<2\pi[/tex] for [tex]x=\dfrac{2\pi}3[/tex] and [tex]x=\dfrac{4\pi}3[/tex]. Extending the domain to account for all real [tex]x[/tex], we have this happening for [tex]x=\dfrac{2\pi}3+2n\pi[/tex] and [tex]\dfrac{4\pi}3+2n\pi[/tex], where [tex]n\in\mathbb Z[/tex].
The second case occurs in [tex]0\le x<2\pi[/tex] when [tex]x=0[/tex], and extending to all reals we have [tex]x=2n\pi[/tex] for [tex]n\in\mathbb Z[/tex], i.e. any even multiple of [tex]\pi[/tex].
