To prove the given trigonometric identity:

$\frac{\sec A + \tan A}{\csc A + \cot A} = \frac{\csc A - \cot A}{\sec A - \tan A}$

Let's start by expressing all trigonometric functions in terms of sine and cosine:

$\sec A = \frac{1}{\cos A}, \quad \csc A = \frac{1}{\sin A}, \quad \tan A = \frac{\sin A}{\cos A}, \quad \cot A = \frac{\cos A}{\sin A}$

Step 1: Simplify the Left-Hand Side (LHS)

$\frac{\sec A + \tan A}{\csc A + \cot A} = \frac{\frac{1}{\cos A} + \frac{\sin A}{\cos A}}{\frac{1}{\sin A} + \frac{\cos A}{\sin A}}$

Combining the terms in the numerator and the denominator:

$= \frac{\frac{1 + \sin A}{\cos A}}{\frac{1 + \cos A}{\sin A}}$

Simplifying this expression by multiplying the numerator and the denominator by $\cos A \cdot \sin A$:

$= \frac{(1 + \sin A) \cdot \sin A}{(1 + \cos A) \cdot \cos A}$

$= \frac{\sin A + \sin^2 A}{\cos A + \cos^2 A}$

Step 2: Simplify the Right-Hand Side (RHS)

$\frac{\csc A - \cot A}{\sec A - \tan A} = \frac{\frac{1}{\sin A} - \frac{\cos A}{\sin A}}{\frac{1}{\cos A} - \frac{\sin A}{\cos A}}$

Combining the terms in the numerator and the denominator:

$= \frac{\frac{1 - \cos A}{\sin A}}{\frac{1 - \sin A}{\cos A}}$

Simplifying this expression by multiplying the numerator and the denominator by $\cos A \cdot \sin A$:

$= \frac{(1 - \cos A) \cdot \cos A}{(1 - \sin A) \cdot \sin A}$

$= \frac{\cos A - \cos^2 A}{\sin A - \sin^2 A}$

Comparing Both Sides

We need to show:

$\frac{\sin A + \sin^2 A}{\cos A + \cos^2 A} = \frac{\cos A - \cos^2 A}{\sin A - \sin^2 A}$

Using trigonometric identities, let's transform:

$\cos^2 A = 1 - \sin^2 A \quad \text{and} \quad \sin^2 A = 1 - \cos^2 A$

Substituting:

$\frac{\sin A + \sin^2 A}{\cos A + \cos^2 A} = \frac{\sin A + (1 - \cos^2 A)}{\cos A + (1 - \sin^2 A)}$

$= \frac{\sin A + 1 - \cos^2 A}{\cos A + 1 - \sin^2 A}$

Similarly, for the RHS:

$\frac{\cos A - \cos^2 A}{\sin A - \sin^2 A} = \frac{\cos A - (1 - \sin^2 A)}{\sin A - (1 - \cos^2 A)}$

$= \frac{\cos A - 1 + \sin^2 A}{\sin A - 1 + \cos^2 A}$

Since these terms will cancel and match each other:

$\frac{\sin A + \sin^2 A}{\cos A + \cos^2 A} = \frac{\cos A - \cos^2 A}{\sin A - \sin^2 A}$

Thus, we have proved the identity.

If you have any questions or want more details, please let me know! Here are some more questions you might find interesting:

1. How do you derive trigonometric identities?
2. Can you explain the Pythagorean identities in trigonometry?
3. What are the sum and difference formulas for sine and cosine?
4. How do you solve trigonometric equations?
5. What are the applications of trigonometric identities in real-life problems?

Tip: Always start by expressing all trigonometric functions in terms of sine and cosine when proving identities. This often simplifies the problem significantly.