Differentiate `tan^(-1)sqrt((1-x^(2))/(1+x^(2)))` with respect to `cos^(-1)x^(2).`

Explanation

To differentiate \( y = \tan^{-1} \left( \sqrt{\frac{1 - x^2}{1 + x^2}} \right) \) with respect to \( \theta = \cos^{-1}(x^2) \), we can follow these steps:

### Step 1: Rewrite the expression
Let \( y = \tan^{-1} \left( \sqrt{\frac{1 - x^2}{1 + x^2}} \right) \).

### Step 2: Substitute \( x^2 \) in terms of \( \theta \)
Since \( \theta = \cos^{-1}(x^2) \), we have:
\[
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