The value of `lim_(xrarrpi)(tan(picos^(2)x))/(sin^(2)(2x))` is equal to

Options

A. 1
B. `pi`
C. `-(pi)/(4)`
D. `(pi)/(2)`

Answer

To solve the limit \( \lim_{x \to \pi} \frac{\tan(\pi \cos^2 x)}{\sin^2(2x)} \), we will follow these steps:

### Step 1: Substitute \( x = \pi \)
First, we substitute \( x = \pi \) into the limit expression:

\[ \tan(\pi \cos^2(\pi)) = \tan(\pi \cdot (-1)^2) = \tan(\pi) = 0 \]
\[ \sin^2(2\pi) = \sin^2(0) = 0 \]

This gives us the indeterminate form \( \frac{0}{0} \).

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that we can take the derivative of the numerator and the denominator:

\[ \lim_{x \to \pi} \frac{\tan(\pi \cos^2 x)}{\sin^2(2x)} = \lim_{x \to \pi} \frac{\frac{d}{dx}[\tan(\pi \cos^2 x)]}{\frac{d}{dx}[\sin^2(2x)]} \]

### Step 3: Differentiate the Numerator
Using the chain rule, the derivative of the numerator is:

\[ \frac{d}{dx}[\tan(\pi \cos^2 x)] = \sec^2(\pi \cos^2 x) \cdot \frac{d}{dx}[\pi \cos^2 x] \]
\[ = \sec^2(\pi \cos^2 x) \cdot (-\pi \cdot 2 \cos x \sin x) = -2\pi \cos x \sin x \sec^2(\pi \cos^2 x) \]

### Step 4: Differentiate the Denominator
The derivative of the denominator is:

\[ \frac{d}{dx}[\sin^2(2x)] = 2 \sin(2x) \cdot \frac{d}{dx}[2x] = 4 \sin(2x) \cos(2x) \]

### Step 5: Substitute the Derivatives Back into the Limit
Now substituting back into the limit gives:

\[ \lim_{x \to \pi} \frac{-2\pi \cos x \sin x \sec^2(\pi \cos^2 x)}{4 \sin(2x) \cos(2x)} \]

### Step 6: Simplify the Limit
We know that \( \sin(2x) = 2 \sin x \cos x \), so we can simplify:

\[ = \lim_{x \to \pi} \frac{-2\pi \cos x \sin x \sec^2(\pi \cos^2 x)}{4 \cdot 2 \sin x \cos x \cos(2x)} \]
\[ = \lim_{x \to \pi} \frac{-\pi \sec^2(\pi \cos^2 x)}{4 \cos(2x)} \]

### Step 7: Evaluate the Limit
Now we substitute \( x = \pi \):

\[ \sec^2(\pi \cos^2(\pi)) = \sec^2(\pi \cdot 1) = \sec^2(\pi) = 1 \]
\[ \cos(2\pi) = 1 \]

Thus, we have:

\[ = \frac{-\pi \cdot 1}{4 \cdot 1} = -\frac{\pi}{4} \]

### Final Answer
Therefore, the value of the limit is:

\[ \boxed{-\frac{\pi}{4}} \]