Solve 2 sin ⁡ x cos ⁡ x = 1 + 3 tan ⁡ x

Note that $\tan x+\cot x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}=\frac{1}{\sin x\cos x}$ (this is something I've filed away from seeing it in a lot of "verify this trig identity" type problems) .
So your equation can be written $2(\tan x+\cot x)=1+3\tan x$, or
$2(\tan x+\frac{1}{\tan x})=1+3\tan x.$
Now that everything is in terms of tangent, you should be able to get it into the form of an equation that's quadratic in $\tan x$ (i.e., has multiples of ${\tan }^{2}x,\tan x,$, and a number) and solve; the values of tanx are nice integers, although you will need inverse trig to get one of the x-values.