Answer
$\{ 0.4636, 0.9828, 3.6052, 4.1244 \}$
Work Step by Step
Step 1. Let $u=tan(x)$; we can rewrite the equation as $4u^2-8u+3=(2u-1)(2u-3)=0$ which gives $u=\frac{3}{2}$ and $u=\frac{1}{2}$
Step 2. For $tan(x)=u=\frac{3}{2}$, we can find the reference angle as $x_0=tan^{-1}(\frac{3}{2})\approx0.9828$
and all x-values in $[0,2\pi)$ as $x\approx0.9828$ and $x=\pi+x_0=4.1244$
Step 3. For $tan(x)=u=\frac{1}{2}$, we can find the reference angle as $x_0=tan^{-1}(\frac{1}{2})\approx0.4636$,
and all x-values in $[0,2\pi)$ as $x\approx0.4636$ and $x=\pi+x_0\approx3.6052$
Step 4. The solutions for the original equation in $[0,2\pi)$ are $\{ 0.4636, 0.9828, 3.6052, 4.1244 \}$
