$\left| {\frac{x}{2} – \frac{\pi }{2}} \right| \le \frac{{3\pi }}{4}$ possible $x$ are

$\begin{array}{l} – \frac{{3\pi }}{4} \le \frac{x}{2} – \frac{\pi }{2} \le \frac{{3\pi }}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, – \frac{\pi }{2},0,\frac{\pi }{2},\pi ,\frac{{3\pi }}{2},2\pi ,\frac{{5\pi }}{2}\\ – \frac{\pi }{4} \le \frac{x}{2} \le \frac{{5\pi }}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sin \frac{x}{2} – \cos \frac{x}{2} = {(\sin \frac{x}{2} – \cos \frac{x}{2})^2}\\ – \frac{\pi }{2} \le x \le \frac{{5\pi }}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,factors\,\sin \frac{x}{2} – \cos \frac{x}{2} = \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\sin \frac{x}{2} – \cos \frac{x}{2} = 1\end{array}$

only circled angle satisfy one of the above equation when $n = 1, 2, 4, 5$