Prove that `tan(pi/16)+2tan(pi/8)+4=cot(pi/6)` .

Explanation

To prove that \( \tan\left(\frac{\pi}{16}\right) + 2\tan\left(\frac{\pi}{8}\right) + 4 = \cot\left(\frac{\pi}{6}\right) \), we will start by evaluating the left-hand side (LHS) and the right-hand side (RHS) separately.

### Step 1: Evaluate the RHS
The right-hand side is \( \cot\left(\frac{\pi}{6}\right) \).
Using the known value:
\[ \cot\left(\frac{\pi}{6}\right) = \frac{1}{\tan\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]
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