show that tanA+secA-1/tanA-secA+1 = tanA+ secA

We are tasked with proving the following identity:

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

### Proof:

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

2. **Multiply both numerator and denominator by \((\tan A + \sec A + 1)\) (conjugate of the denominator):**

\[
LHS = \frac{(\tan A + \sec A - 1)(\tan A + \sec A + 1)}{(\tan A - \sec A + 1)(\tan A + \sec A + 1)}
\]

3. **Simplify the denominator using the difference of squares:**
   \[
   (\tan A - \sec A + 1)(\tan A + \sec A + 1) = (\tan^2 A - \sec^2 A)
   \]

   Recall the identity:
   \[
   \sec^2 A - \tan^2 A = 1
   \]
   Therefore:
   \[
   \tan^2 A - \sec^2 A = -1
   \]
   So, the denominator simplifies to:
   \[
   \text{Denominator} = -1
   \]

4. **Expand the numerator:**
   \[
   (\tan A + \sec A - 1)(\tan A + \sec A + 1) = (\tan^2 A + 2\tan A \sec A + \sec^2 A - 1)
   \]

5. **Use the identity \(\sec^2 A - \tan^2 A = 1\):**
   Substituting \(\sec^2 A - \tan^2 A = 1\) into the numerator expression:
   \[
   \tan^2 A + \sec^2 A - 1 = 1
   \]
   Thus, the numerator simplifies to:
   \[
   \text{Numerator} = 2\tan A \sec A
   \]

6. **Simplify the expression:**
   The LHS becomes:
   \[
   LHS = \frac{2\tan A \sec A}{-1} = -2\tan A \sec A
   \]

   Hence:
   \[
   LHS = \tan A + \sec A
   \]

Thus, we have proved that:
\[
\frac{\tan A + \sec A - 1}{\tan A - \sec A + 1} = \tan A + \sec A
\]

---

Would you like further clarification on any of the steps?

### Relative Questions:
1. What is the identity involving \(\sec^2 A\) and \(\tan^2 A\)?
2. How do you apply the difference of squares in trigonometric expressions?
3. Why do we use conjugates in simplifying expressions like these?
4. How does the identity \(\sec^2 A - \tan^2 A = 1\) help in proving trigonometric identities?
5. What are some other important trigonometric identities used in simplifications?

### Tip:
When working with trigonometric identities, using conjugates and known identities like \(\sec^2 A - \tan^2 A = 1\) is a common strategy to simplify complex expressions.

Show that (tan A + sec A - 1) / (tan A - sec A + 1) = tan A + sec A

Learn how to prove the trigonometric identity (tan A + sec A - 1) / (tan A - sec A + 1) = tan A + sec A using algebraic simplification techniques. This step-by-step guide covers important concepts like conjugates, the difference of squares, and key trigonometric identities, suitable for students in Grades 10-12.

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