Prove the following identity, where the angles involved are acute angles for which the expressions are defined.
(x) `((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cotA))^2=tan^2A`

Explanation

To prove the identity \(\frac{1 + \tan^2 A}{1 + \cot^2 A} = \left(\frac{1 - \tan A}{1 - \cot A}\right)^2 = \tan^2 A\), we will break it down into two parts and prove each part step by step.

### Step 1: Prove \(\frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A\)

1. **Start with the left-hand side (LHS)**:
\[ \text{LHS} = \frac{1 + \tan^2 A}{1 + \cot^2 A} \]
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