$\sec \theta =\frac{a^2+b^2}{a^2-b^2}$, where $a,b,\in R$, gives real values of $\theta$ if and only if

Clearly,

$a^2+b^2\ge a^2-b^2$ for all $\left|\right|a|\ne |b\mid \ne 0$,

$\Rightarrow \frac{a^2+b^2}{a^2-b^2}\ge 1$ or. $\frac{a^2+b^2}{a^2-b^2}\le -1$

$\therefore \sec \theta =\frac{a^2+b^2}{a^2-b^2}$ is meaningful.

Thus, $\sec \theta =\frac{a^2+b^2}{a^2-b^2}$ gives real values of $\theta$ if and only $\left|a\right|\ne \left|b\right|\ne 0$