Answer
$x=\frac{\pi}{3}$ and $x=\frac{5\pi}{3}$
Work Step by Step
$cos~x+sin~x~tan~x=2$
$cos~x+sin~x~\frac{sin~x}{cos~x}=2$
$\frac{cos~x(cos~x)}{cos~x}+\frac{sin^2x}{cos~x}=2$
$\frac{cos^2x+sin^2x}{cos~x}=2$
$\frac{1}{cos~x}=2$
$cos~x=\frac{1}{2}$
The solutions in the interval $[0,2\pi)$ are:
$x=\frac{\pi}{3}$ and $x=\frac{5\pi}{3}$
