As a start...
$\displaystyle \begin{align*} \tan{(x)} + \sec{(x)} &= \sqrt{3} \\ \frac{\sin{(x)}}{\cos{(x)}} + \frac{1}{\cos{(x)}} &= \sqrt{3} \\ \frac{\sin{(x)} + 1}{\cos{(x)}} &= \sqrt{3} \\ \sin{(x)} + 1 &= \sqrt{3}\cos{(x)} \end{align*}$
Now you need some way to convert everything to sines or cosines. Squaring and applying the Pythagorean Identity would work, you just need to be careful for extraneous solutions.