Solve `tan^(-1) x + cot^(-1) (-|x|) = 2 tan^(-1) 6x`

Explanation

To solve the equation \( \tan^{-1} x + \cot^{-1} (-|x|) = 2 \tan^{-1} (6x) \), we will break it down step by step.

### Step 1: Rewrite the cotangent inverse
We start with the equation:
\[ \tan^{-1} x + \cot^{-1} (-|x|) = 2 \tan^{-1} (6x) \]
Since \( \cot^{-1}(-|x|) = \pi - \cot^{-1}(|x|) \) for \( x < 0 \) and \( \cot^{-1}(-|x|) = \pi - \cot^{-1}(x) \) for \( x > 0 \), we will consider both cases.
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