If $2\sin \frac{A}{2}=\sqrt{1+\sin A}+\sqrt{1-\sin A}then\frac{A}{2}liesbetween$

$$\begin{gathered} We have \sqrt{1+\sin A}=\left|\cos \frac{A}{2}+\sin \frac{A}{2}\right|=\left|C+S\right|\left(say\right)\to \left(1\right) \\ and \sqrt{1-\sin A}=\left|\cos \frac{A}{2}-\sin \frac{A}{2}\right|=\left|C-S\right|\left(say\right)\to \left(2\right) \\ We can have \sqrt{1+\sin A}+\sqrt{1-\sin A}=2\sin \frac{A}{2} if C+S>0 and C-S<0 \\ Now C+S>0\Rightarrow -\frac{\mathrm{\pi }}{4}<\frac{A}{2}<\frac{3\mathrm{\pi }}{4}, and C-S<0\Rightarrow \frac{\mathrm{\pi }}{4}<\frac{A}{2}<\frac{5\mathrm{\pi }}{4} \\ \Rightarrow \frac{\mathrm{\pi }}{4}<\frac{A}{2}<\frac{3\mathrm{\pi }}{4} \\ \Rightarrow 2n\mathrm{\pi }+\frac{\mathrm{\pi }}{4}<\frac{A}{2}<2n\mathrm{\pi }+\frac{3\mathrm{\pi }}{4} \end{gathered}$$