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!