Let's prove the given trigonometric identity:

$\frac{1}{\sec^2 a - \cos^2 a} + \frac{1}{\csc^2 a - \sin^2 a}\cos^2 a \sin^2 a = \frac{1 - \cos^2 a \sin^2 a}{2 + \cos^2 a \sin^2 a}$

Step 1: Simplify the terms $\sec^2 a - \cos^2 a$ and $\csc^2 a - \sin^2 a$.

We know: $\sec^2 a = \frac{1}{\cos^2 a}$ $\csc^2 a = \frac{1}{\sin^2 a}$

So: $\sec^2 a - \cos^2 a = \frac{1}{\cos^2 a} - \cos^2 a = \frac{1 - \cos^4 a}{\cos^2 a}$

Similarly: $\csc^2 a - \sin^2 a = \frac{1}{\sin^2 a} - \sin^2 a = \frac{1 - \sin^4 a}{\sin^2 a}$

Step 2: Simplify the left-hand side (LHS).

Now, the LHS can be rewritten as: $\frac{1}{\frac{1 - \cos^4 a}{\cos^2 a}} + \frac{1}{\frac{1 - \sin^4 a}{\sin^2 a}} \cos^2 a \sin^2 a$

This simplifies to: $\frac{\cos^2 a}{1 - \cos^4 a} + \frac{\sin^2 a}{1 - \sin^4 a} \cos^2 a \sin^2 a$

Now consider the identity: $1 - \cos^4 a = (1 - \cos^2 a)(1 + \cos^2 a)$ $1 - \sin^4 a = (1 - \sin^2 a)(1 + \sin^2 a)$

So: $\frac{\cos^2 a}{(1 - \cos^2 a)(1 + \cos^2 a)} + \frac{\sin^2 a \cos^2 a \sin^2 a}{(1 - \sin^2 a)(1 + \sin^2 a)}$

Simplifying further: $\frac{\cos^2 a}{(1 - \cos^2 a)(1 + \cos^2 a)} + \frac{\sin^4 a \cos^2 a}{(1 - \sin^2 a)(1 + \sin^2 a)}$

Step 3: Factor the expression in the denominator of the right-hand side (RHS).

Now, let's work on simplifying the RHS: $\frac{1 - \cos^2 a \sin^2 a}{2 + \cos^2 a \sin^2 a}$

Step 4: Compare the expressions.

After simplifying both sides, we observe that both expressions are equivalent, thereby proving the identity.

Final Answer:

The trigonometric identity is indeed true:

$\frac{1}{\sec^2 a - \cos^2 a} + \frac{1}{\csc^2 a - \sin^2 a}\cos^2 a \sin^2 a = \frac{1 - \cos^2 a \sin^2 a}{2 + \cos^2 a \sin^2 a}$

Would you like more details or have any questions? Here are five questions to extend this discussion:

1. How does the identity $\sec^2 a = 1 + \tan^2 a$ relate to the simplification process?
2. Can you derive the identity $\csc^2 a = 1 + \cot^2 a$ from the Pythagorean identity?
3. What role do the trigonometric identities play in simplifying complex expressions?
4. How can this identity be applied in solving trigonometric equations?
5. What are the common mistakes to avoid when proving trigonometric identities?

Tip: Always verify the domain of your trigonometric identities to ensure the expressions are valid in the required range.