The problem asks to find the limit of the given function as $x \to \frac{\pi}{2}$:

$\lim_{x \to \frac{\pi}{2}} \frac{\cos^2(x)}{\left(\frac{\pi}{2} - x\right)^2 \sec^2\left(x + \frac{\pi}{4}\right)}$

Step-by-step solution:

1. Analyze the behavior of the cosine function:

   As $x \to \frac{\pi}{2}$, we know that $\cos(x) \to 0$. Hence, $\cos^2(x)$ approaches 0 as well.

2. Simplify $\sec^2\left(x + \frac{\pi}{4}\right)$:

   The secant function is the reciprocal of cosine, i.e., $\sec(x) = \frac{1}{\cos(x)}$. Therefore: $\sec^2\left(x + \frac{\pi}{4}\right) = \frac{1}{\cos^2\left(x + \frac{\pi}{4}\right)}$ We also know that $x + \frac{\pi}{4}$ approaches $\frac{3\pi}{4}$ as $x \to \frac{\pi}{2}$. Using the identity $\cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}}$, we get: $\sec^2\left(x + \frac{\pi}{4}\right) \to 2$

3. Examine the limit of $\frac{\pi}{2} - x$:

   As $x \to \frac{\pi}{2}$, the term $\frac{\pi}{2} - x \to 0$, so $\left(\frac{\pi}{2} - x\right)^2 \to 0$.

4. Determine the overall limit:

   The numerator $\cos^2(x)$ and the denominator $\left(\frac{\pi}{2} - x\right)^2$ both tend to 0. This suggests applying L'Hopital's Rule for the indeterminate form $\frac{0}{0}$.

   • Differentiate the numerator: $\frac{d}{dx} \cos^2(x) = 2\cos(x)(-\sin(x)) = -2\cos(x)\sin(x) = -\sin(2x)$

   • Differentiate the denominator: $\frac{d}{dx} \left[\left(\frac{\pi}{2} - x\right)^2 \sec^2\left(x + \frac{\pi}{4}\right)\right]$

     This differentiation involves the product rule, but it simplifies due to the behavior of $\sec^2\left(x + \frac{\pi}{4}\right)$ being finite and the first term dominating.

5. Final result: The limit simplifies to $2$, since the terms involving $\sec^2\left(x + \frac{\pi}{4}\right)$ approach constants, and the main rate of approach is determined by the first term.

Thus, the value of the limit is:

$\boxed{2}$

Would you like a detailed breakdown of L'Hopital's Rule in this context or any further clarification?

Here are 5 related questions:

1. What is L'Hopital's Rule and how is it applied in limits?
2. How can we handle limits involving trigonometric functions near their critical points?
3. Why does $\sec(x)$ behave differently from $\cos(x)$ as $x \to \frac{\pi}{2}$?
4. How does the chain rule affect the differentiation of composite functions like $\sec^2(x + \frac{\pi}{4})$?
5. What are other methods to solve indeterminate forms besides L'Hopital's Rule?

Tip: When solving limits involving trigonometric functions, try using known trigonometric limits and identities to simplify before applying advanced techniques like L'Hopital's Rule.