Solve: `tan4theta=-cot(pi/6+theta)`

Explanation

To solve the equation \( \tan 4\theta = -\cot\left(\frac{\pi}{6} + \theta\right) \), we can follow these steps:

### Step 1: Rewrite the cotangent
We know that \( \cot x = -\tan\left(\frac{\pi}{2} + x\right) \). Therefore, we can rewrite the equation as:
\[ \tan 4\theta = -\cot\left(\frac{\pi}{6} + \theta\right) = \tan\left(\frac{\pi}{2} + \left(\frac{\pi}{6} + \theta\right)\right) \]
This simplifies to:
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