Answer
$\displaystyle \cos^{-1}(\frac{1}{a})$
Work Step by Step
Inverse of Secant:
$y=\sec^{-1}x$ or $y=$ arcsec $x$ means that
$x=\sec y$, for $0\leq y\leq\pi, y\displaystyle \neq\frac{\pi}{2}$.
so
$y=\sec^{-1}a$ means that $a=\sec y,$
... $\sec y$ and $\cos y$ are reciprocal ...
$ a=\displaystyle \frac{1}{\cos y}\qquad /\displaystyle \times\frac{\cos y}{a}$
$\displaystyle \cos y=\frac{1}{a}$
By definition of lnverse Cosine Function
$y=\cos^{-1}x$ or $y=$ arccos $x$ means that $x=\cos y$, for $ 0\leq y\leq\pi$,
so
$\displaystyle \cos y=\frac{1}{a}$ means that $y=\displaystyle \cos^{-1}(\frac{1}{a})$
Calculating $y=\sec^{-1}a$, we calculate $\displaystyle \cos^{-1}(\frac{1}{a})$