The numerical value of `tan(pi/3)+2tan((2pi)/3)+4tan((4pi)/3)+8tan((8pi)/3)` is equal to (A) `-5sqrt(3)` (B) `-5/(sqrt3)` (C) `5sqrt(3)` (D) `5/(sqrt3)`

Explanation

To solve the expression \( \tan\left(\frac{\pi}{3}\right) + 2\tan\left(\frac{2\pi}{3}\right) + 4\tan\left(\frac{4\pi}{3}\right) + 8\tan\left(\frac{8\pi}{3}\right) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \tan\left(\frac{\pi}{3}\right) \)
The angle \( \frac{\pi}{3} \) is in the first quadrant where tangent is positive. We know:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

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