Principal solution of given trigonometric equation Find the principal solution of the following equation:
$\sin ^2x+2\tan ^2x+\frac{4}{\sqrt3} \tan x -\sin x+\frac{11}{12}=0$
I tried it by converting both $\sin x$ and $\tan x$ into half angles ( of $\tan x/2 $) but it is resulting in ugly equation. Could someone suggest a smarter approach without using half angles?

On some manipulation of the $LHS$ of the equation, it can be shown that it is the same as 
$(sin(x) - \frac{1}{2})^2 + 2(tan(x) + \frac{1}{\sqrt(3)})^2$. And since squares of real numbers are always non-negative, for the sum of the 2 terms to be 0, both of them have to be simultaneously 0.
And both of them are simultaneously zero at $x = 2n\pi + \frac{5\pi}{6}$ for $n \in Z$.
The principal solution is thus $\frac{5\pi}{6}$. 
