# The numerical value of $\tan\large\frac{\pi}{3}$$+2\tan\large\frac{2\pi}{3}$$+4\tan\large\frac{4\pi}{3}$$+8\tan\large\frac{8\pi}{3} is equal to (a)\;-5\sqrt 3\qquad(b)\;\large\frac{-5}{\sqrt 3}\qquad$$(c)\;5\sqrt 3\qquad(d)\;\large\frac{5}{\sqrt 3}$

$\tan\large\frac{\pi}{3}$$+2\tan\large\frac{2\pi}{3}$$+4\tan\large\frac{4\pi}{3}$$+8\tan\large\frac{8\pi}{3} \Rightarrow \tan\large\frac{\pi}{3}+$$2\tan(\pi-\large\frac{\pi}{3})$$+4\tan(\pi+\large\frac{\pi}{3})$$+8\tan(3\pi-\large\frac{\pi}{3})$
$\Rightarrow \tan\large\frac{\pi}{3}$$-2\tan\large\frac{\pi}{3}$$+4\tan\large\frac{\pi}{3}$$-8\tan\large\frac{\pi}{3} \Rightarrow -5\tan\large\frac{\pi}{3} \tan\large\frac{\pi}{3}$$=\sqrt 3$
$\Rightarrow -5\sqrt 3$
Hence (a) is the correct answer.