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The equation you've provided is:

\[
1 + \tan^2(x) = \sec^2(x)
\]

This identity is a well-known trigonometric identity. Here's how it is derived:

### Derivation:

1. **Start with the Pythagorean Identity**:
   \[
   \sin^2(x) + \cos^2(x) = 1
   \]
   
2. **Divide the entire identity by \(\cos^2(x)\)**:
   \[
   \frac{\sin^2(x)}{\cos^2(x)} + \frac{\cos^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)}
   \]
   
3. **Simplify each term**:
   - \(\frac{\sin^2(x)}{\cos^2(x)} = \tan^2(x)\)
   - \(\frac{\cos^2(x)}{\cos^2(x)} = 1\)
   - \(\frac{1}{\cos^2(x)} = \sec^2(x)\)

   So the equation becomes:
   \[
   \tan^2(x) + 1 = \sec^2(x)
   \]

4. **Rearrange** to match your original equation:
   \[
   1 + \tan^2(x) = \sec^2(x)
   \]

This shows how the identity is derived. It's a fundamental trigonometric identity that is very useful in calculus and trigonometric equations.

Would you like more details or have any further questions?

Here are 5 related questions you might find interesting:
1. How can this identity be used to solve trigonometric equations?
2. What is the geometric interpretation of this identity on the unit circle?
3. How does this identity relate to the Pythagorean Theorem?
4. Can this identity be extended to functions other than sine and cosine?
5. What are the analogous identities for other trigonometric functions like cotangent and cosecant?

**Tip:** When dealing with trigonometric identities, try to remember the fundamental identities like the Pythagorean identities, as they are the basis for deriving many other identities.

Explore the derivation and application of the trigonometric identity 1 + tan^2(x) = sec^2(x). Learn how this identity is derived from the Pythagorean identity and its significance in calculus and trigonometry. Suitable for advanced high school students and anyone interested in trigonometric equations.

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