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The problem in the image asks to "Demonstrar las siguientes identidades," and the specific identity to be demonstrated is:

\[
\sec^2(x) + \csc^2(x) = \sec^2(x) \cdot \csc^2(x)
\]

### Step-by-Step Solution

1. **Understand the Identity:**
   The identity we need to prove is:
   \[
   \sec^2(x) + \csc^2(x) = \sec^2(x) \cdot \csc^2(x)
   \]
   where \(\sec(x) = \frac{1}{\cos(x)}\) and \(\csc(x) = \frac{1}{\sin(x)}\).

2. **Rewrite the Identity in Terms of Sine and Cosine:**
   Substitute \(\sec(x)\) and \(\csc(x)\) with their respective definitions:
   \[
   \frac{1}{\cos^2(x)} + \frac{1}{\sin^2(x)} = \frac{1}{\cos^2(x)} \cdot \frac{1}{\sin^2(x)}
   \]

3. **Simplify the Right Side:**
   The right side of the equation can be simplified as follows:
   \[
   \frac{1}{\cos^2(x) \sin^2(x)}
   \]

4. **Find a Common Denominator on the Left Side:**
   The left side requires a common denominator to combine the two fractions:
   \[
   \frac{\sin^2(x) + \cos^2(x)}{\cos^2(x) \sin^2(x)}
   \]

5. **Apply the Pythagorean Identity:**
   The Pythagorean identity states that \(\sin^2(x) + \cos^2(x) = 1\). So, substitute \(1\) for \(\sin^2(x) + \cos^2(x)\):
   \[
   \frac{1}{\cos^2(x) \sin^2(x)}
   \]

6. **Compare Both Sides:**
   Now both sides of the equation are equal:
   \[
   \frac{1}{\cos^2(x) \sin^2(x)} = \frac{1}{\cos^2(x) \sin^2(x)}
   \]
   Hence, the identity is proven to be true.

Would you like to dive deeper into any of these steps or have any questions? Here are some related topics you might find interesting:

1. How to prove other trigonometric identities.
2. The use of the Pythagorean identities in trigonometry.
3. Deriving other identities from \(\sec(x)\) and \(\csc(x)\).
4. The relationship between \(\sec(x)\), \(\csc(x)\), and \(\tan(x)\).
5. Applications of trigonometric identities in calculus.
6. Simplifying complex trigonometric expressions.
7. Solving trigonometric equations.
8. Graphical interpretation of trigonometric identities.

**Tip:** When proving trigonometric identities, always look for opportunities to apply fundamental identities like the Pythagorean identity, as they often simplify the problem significantly.

Learn how to prove the trigonometric identity sec^2(x) + csc^2(x) = sec^2(x) * csc^2(x) step-by-step with detailed explanations. This problem involves applying the Pythagorean identity and understanding trigonometric identities. Suitable for high school students.

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