#### Answer

$\frac{\pi}{3}$, $\frac{5\pi}{3}$

#### Work Step by Step

$2\sec x+1=\sec x+3$
$\sec x+1=3$
$\sec x=2$
$\cos x=\frac{1}{2}$
Referring to the unit circle (and remembering that cosine is positive in Quadrants I and IV), we see that the only solutions in $[0, 2\pi)$ are $\frac{\pi}{3}$ and $\frac{5\pi}{3}$.