Eliminate the parameter to find the cartesian equation of the curve
\[
x=5 \sec \theta, \quad y=5 \tan \theta, \quad-\frac{\pi}{2}< \theta< \frac{\pi}{2} .
\]

The equation of the curve is:
\[
x=
\]

Step 1 ：First, we express $\sec \theta$ and $\tan \theta$ in terms of $x$ and $y$ respectively.

Step 2 ：From $x=5 \sec \theta$, we get $\sec \theta = \frac{x}{5}$.

Step 3 ：From $y=5 \tan \theta$, we get $\tan \theta = \frac{y}{5}$.

Step 4 ：Now, we substitute these expressions into the identity $\sec^2 \theta = 1 + \tan^2 \theta$ to get an equation in terms of $x$ and $y$ only.

Step 5 ：This gives us $\left(\frac{x}{5}\right)^2 = 1 + \left(\frac{y}{5}\right)^2$.

Step 6 ：Simplifying this equation gives us the cartesian equation of the curve.

Step 7 ：The cartesian equation of the curve is $\boxed{x^2 = 25 + y^2}$.