Minimum value

We can use trigonometry identities to re-write $13\sec\theta-9\sin\theta\tan\theta$ as:

$\frac{13}{\cos\theta}-\frac{9\sin^2\theta}{\cos\theta}=\frac{13-9\sin^2\theta}{\cos\theta}=\frac{4}{\cos\theta}+9\cos\theta$

Since $\frac{-\pi}{2}<\theta<\frac{\pi}{2}$ , then $1\geq\cos\theta>0$ , so we can apply AM-GM inequality to get $\frac{4}{\cos\theta}+9\cos\theta\geq2\sqrt{4\times9}=12$ . Equality happens when $\cos\theta=\frac{2}{3}$ .