The limit expression you provided is:

$\lim_{\theta \to \frac{\pi}{4}} \frac{\cos(2\theta)}{\cos(\theta) \sin(\theta)}$

Step-by-step Solution:

1. Apply the trigonometric identity:
   Recall that $\cos(2\theta) = 2\cos^2(\theta) - 1$.

2. Evaluate at $\theta = \frac{\pi}{4}$:

   • $\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$
   • $\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$

   So the denominator becomes: $\cos\left(\frac{\pi}{4}\right) \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2}$

   The numerator is: $\cos\left(2 \times \frac{\pi}{4}\right) = \cos\left(\frac{\pi}{2}\right) = 0$

3. Limit calculation:
   The numerator $\cos(2\theta)$ tends to 0 as $\theta \to \frac{\pi}{4}$, while the denominator $\cos(\theta)\sin(\theta)$ approaches $\frac{1}{2}$. Therefore:

   $\lim_{\theta \to \frac{\pi}{4}} \frac{\cos(2\theta)}{\cos(\theta) \sin(\theta)} = \frac{0}{\frac{1}{2}} = 0$

Final Answer:

$\boxed{0}$

Would you like more details on any specific step, or further examples?

Related Questions:

1. How does the double angle identity for cosine work?
2. What are the limits of other trigonometric functions as $\theta \to \frac{\pi}{4}$?
3. How do you apply L'Hopital's Rule for indeterminate forms in limits?
4. What are other methods to simplify trigonometric limits?
5. Can this limit be solved using Taylor series approximations?

Tip:

Whenever you encounter trigonometric limits, always check if identities like double angle formulas or factorizations can simplify the expression!