Example 7 Given the function $y=\sqrt{1+\sin x}+\sqrt{1-\sin x}$, find the maximum value of $y$.
(Singapore Competition Question)
Let $u=\sqrt{1+\sin x}, v=\sqrt{1-\sin x}$, then we have the line equation: $u+v-y=0$, and the circle equation: $u^{2}+v^{2}=2$.
From the given, we know that the line and the circle have a common point $(u, v)$, thus
$\frac{|y|}{\sqrt{2}} \leqslant \sqrt{2}$, i.e., $|y| \leqslant 2$, equality holds if and only if $x=0$, hence $y_{\text {max }}=2$.