1. The area of the inscribed square in the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$ is
$-1 . \frac{450}{17}$.
By the symmetry of the ellipse, we know that the sides of the inscribed square should be parallel to the coordinate axes, and the center is at the origin. Therefore, the equations of the diagonals are $y= \pm x$.
Substituting into the ellipse equation, we get
$$
x^{2}\left(\frac{1}{5^{2}}+\frac{1}{3^{2}}\right)=1 \Rightarrow S=(2 x)^{2}=\frac{450}{17} \text {. }
$$