Answer
$sec~x = \pm \frac{\sqrt{1 - sin^2~x}}{1 - sin^2~x}$
Work Step by Step
We can find an expression for $cos~x$ in terms of $sin~x$:
$sin^2~x+cos^2~x = 1$
$cos^2~x = 1 - sin^2~x$
$cos~x = \pm~\sqrt{1 - sin^2~x}$
We can write $sec~x$ in terms of $sin~x$:
$sec~x = \frac{1}{cos~x}$
$sec~x = \frac{1}{\pm~\sqrt{1 - sin^2~x}}$
$sec~x = \frac{1}{\pm~\sqrt{1 - sin^2~x}}~\frac{\sqrt{1 - sin^2~x}}{\sqrt{1 - sin^2~x}}$
$sec~x = \pm \frac{\sqrt{1 - sin^2~x}}{1 - sin^2~x}$
