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To prove the identity \( \sec^2 x + \csc^2 x = \sec^2 x \csc^2 x \), let's break down the steps systematically.

### Given Identity:
\[
\sec^2 x + \csc^2 x = \sec^2 x \csc^2 x
\]

### Steps:

1. **Rewrite secant and cosecant** in terms of sine and cosine:
   \[
   \sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}
   \]
   Thus:
   \[
   \sec^2 x = \frac{1}{\cos^2 x}, \quad \csc^2 x = \frac{1}{\sin^2 x}
   \]

2. **Substitute** these into the left-hand side (LHS):
   \[
   \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x}
   \]
   To simplify this, take the common denominator:
   \[
   \frac{\sin^2 x + \cos^2 x}{\cos^2 x \sin^2 x}
   \]

3. **Simplify** using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \):
   \[
   \frac{1}{\cos^2 x \sin^2 x}
   \]

4. **Compare** the result with the right-hand side (RHS):
   The right-hand side is \( \sec^2 x \csc^2 x \), which can also be written as:
   \[
   \frac{1}{\cos^2 x} \times \frac{1}{\sin^2 x} = \frac{1}{\cos^2 x \sin^2 x}
   \]

Since both sides are equal, the identity is **proved**:
\[
\sec^2 x + \csc^2 x = \sec^2 x \csc^2 x
\]

### Would you like further details or clarification on any step? 

Here are 5 related questions to explore further:

1. How do you derive the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \)?
2. What are other important trigonometric identities similar to the one used here?
3. How would you prove the identity \( \tan^2 x + 1 = \sec^2 x \)?
4. Can you solve this identity using a different method?
5. How do you simplify trigonometric expressions involving multiple identities?

**Tip:** Always remember to convert complicated trigonometric expressions into sine and cosine to simplify proofs!

Prove the identity: sec^2 x + csc^2 x = sec^2 x csc^2 x

Learn how to prove the trigonometric identity sec^2 x + csc^2 x = sec^2 x csc^2 x step by step. This problem covers key trigonometric identities and simplification techniques. Suitable for high school students in Grades 9-12.

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