(tan A)/(1 - cot A) + (cot A)/(1 - tan A) = cosec(A) * sec(A) + 1

To prove the identity

\[
\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} = \csc A \cdot \sec A + 1
\]

### Step 1: Express \(\tan A\) and \(\cot A\) in terms of \(\sin A\) and \(\cos A\).
- \(\tan A = \frac{\sin A}{\cos A}\)
- \(\cot A = \frac{\cos A}{\sin A}\)

Substituting these into the left-hand side (LHS):

\[
\text{LHS} = \frac{\frac{\sin A}{\cos A}}{1 - \frac{\cos A}{\sin A}} + \frac{\frac{\cos A}{\sin A}}{1 - \frac{\sin A}{\cos A}}
\]

### Step 2: Simplify the expressions inside the fractions.

For the first fraction:

\[
\frac{\sin A / \cos A}{1 - \cos A / \sin A} = \frac{\sin A / \cos A}{\frac{\sin A - \cos A}{\sin A}} = \frac{\sin^2 A}{\cos A (\sin A - \cos A)}
\]

For the second fraction:

\[
\frac{\cos A / \sin A}{1 - \sin A / \cos A} = \frac{\cos A / \sin A}{\frac{\cos A - \sin A}{\cos A}} = \frac{\cos^2 A}{\sin A (\cos A - \sin A)}
\]

So, the LHS becomes:

\[
\text{LHS} = \frac{\sin^2 A}{\cos A (\sin A - \cos A)} + \frac{\cos^2 A}{\sin A (\cos A - \sin A)}
\]

### Step 3: Combine the two fractions.

Since both terms have a common denominator, \(\cos A (\sin A - \cos A)\) and \(\sin A (\cos A - \sin A)\), we get:

\[
\text{LHS} = \frac{\sin^3 A - \cos^3 A}{\cos A \sin A (\cos A - \sin A)}
\]

### Step 4: Factor and simplify the expression.

The numerator can be factored as:

\[
\sin^3 A - \cos^3 A = (\sin A - \cos A)(\sin^2 A + \sin A \cos A + \cos^2 A)
\]

The denominator simplifies as well. Therefore:

\[
\text{LHS} = \frac{(\sin A - \cos A)(\sin^2 A + \sin A \cos A + \cos^2 A)}{\sin A \cos A (\cos A - \sin A)}
\]

Notice that \(\sin A - \cos A\) and \(\cos A - \sin A\) are negatives of each other, so this cancels out and simplifies to:

\[
\text{LHS} = \frac{\sin^2 A + \cos^2 A + \sin A \cos A}{\sin A \cos A}
\]

Using the identity \(\sin^2 A + \cos^2 A = 1\), this simplifies to:

\[
\text{LHS} = \frac{1 + \sin A \cos A}{\sin A \cos A} = \csc A \cdot \sec A + 1
\]

This matches the right-hand side (RHS) of the equation, proving the identity.

### Conclusion:
\[
\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} = \csc A \cdot \sec A + 1
\]

This identity is verified.

Do you have any questions or need further details?

### Related Questions:

1. How can we derive \(\csc A\) and \(\sec A\) in terms of \(\tan A\) and \(\cot A\)?
2. What is the significance of trigonometric identities in solving equations?
3. How can the sum-to-product identities simplify trigonometric expressions?
4. Can we derive the identity using a different method?
5. What are the practical applications of trigonometric identities?

**Tip:** When dealing with trigonometric identities, always start by expressing all terms in sine and cosine. This often simplifies the expression.

Learn how to prove the trigonometric identity (tan A)/(1 - cot A) + (cot A)/(1 - tan A) = cosec(A) * sec(A) + 1 step-by-step with detailed explanations and mathematical formulas. This problem involves advanced trigonometric concepts and identities, suitable for advanced high school students.

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