Trigonometric identity

Prove equality
$$
\tan(\frac{3\pi}{11})+4\sin(\frac{2\pi}{11})=\sqrt{11}.
$$

This is the source. I didn't find a better way, and my method is a bit surprising though also takes quite some computation.


Proof:
Define complex number
$$
q=\exp{(\frac{j\pi}{11})}.
$$

Then the first term becomes $\frac{q^3-q^{-3}}{j(q^3+q^{-3})}$, and the second $4\frac{q^2-q^{-2}}{2j}$. Now translate the desired equality to another that $q$ is known to satisfy:

$$
q^{10}-q^9+\ldots+q^2-q+1=0.
$$

Q.E.D.