We are given the equation to prove:

$\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} = 1 + \tan A + \cot A$

Steps to Prove the Identity:

1. Convert $\cot A$ to $\tan A$: We know that $\cot A = \frac{1}{\tan A}$. Substituting this into the equation:

   $\frac{\tan A}{1 - \frac{1}{\tan A}} + \frac{\frac{1}{\tan A}}{1 - \tan A}$

2. Simplify the terms: Simplify the denominator in the first fraction:

   $\frac{\tan A}{\frac{\tan A - 1}{\tan A}} = \frac{\tan A \cdot \tan A}{\tan A - 1} = \frac{\tan^2 A}{\tan A - 1}$

   Simplify the second fraction:

   $\frac{\frac{1}{\tan A}}{1 - \tan A} = \frac{1}{\tan A (1 - \tan A)}$

   Now the equation becomes:

   $\frac{\tan^2 A}{\tan A - 1} + \frac{1}{\tan A (1 - \tan A)}$

3. Get common denominators: The common denominator of these two fractions will be $(\tan A - 1)(1 - \tan A)$. So we rewrite the fractions:

   $\frac{\tan^2 A}{\tan A - 1} + \frac{1}{1 - \tan A}$

4. Combine fractions: Add the two fractions with the common denominator:

   $\frac{\tan^2 A + 1}{(1 - \tan A)(\tan A - 1)}$

   Since $(1 - \tan A)(\tan A - 1) = -(\tan A - 1)^2$, the equation becomes:

   $\frac{\tan^2 A + 1}{-(\tan A - 1)^2}$

5. Use trigonometric identities: Now use the identity $\tan^2 A + 1 = \sec^2 A$. So the expression simplifies to:

   $\frac{\sec^2 A}{-(\tan A - 1)^2}$

   This should simplify further to give us the right-hand side, but it appears there's a sign issue in how we've proceeded. Let me review the algebra and approach it differently