If `cos^(-1).(6x)/(1 + 9x^(2)) = -(pi)/(2) + 2 tan^(-1) 3x`, then find the values of x

Explanation

To solve the equation \( \cos^{-1}\left(\frac{6x}{1 + 9x^2}\right) = -\frac{\pi}{2} + 2 \tan^{-1}(3x) \), we will follow these steps:

### Step 1: Rewrite the equation using the identity for cosine inverse
We know that:
\[ \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x) \]
Thus, we can rewrite the left-hand side:
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