`|(3-sec^2 x)_(max) -( 4+tan^2 y)_(min)|` equals

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem \( |(3 - \sec^2 x)_{\text{max}} - (4 + \tan^2 y)_{\text{min}}| \), we will follow these steps:

### Step 1: Analyze \( (3 - \sec^2 x)_{\text{max}} \)
To maximize \( 3 - \sec^2 x \), we need to minimize \( \sec^2 x \). The function \( \sec^2 x \) is defined as \( \sec^2 x = 1 + \tan^2 x \), which is always greater than or equal to 1. Therefore, the minimum value of \( \sec^2 x \) is 1.

Substituting this value:
\[ (3 - \sec^2 x)_{\text{max}} = 3 - 1 = 2 \]

### Step 2: Analyze \( (4 + \tan^2 y)_{\text{min}} \)
To minimize \( 4 + \tan^2 y \), we need to minimize \( \tan^2 y \). The function \( \tan^2 y \) is also always greater than or equal to 0. Therefore, the minimum value of \( \tan^2 y \) is 0.

Substituting this value:
\[ (4 + \tan^2 y)_{\text{min}} = 4 + 0 = 4 \]

### Step 3: Calculate the expression
Now we substitute the values we found into the original expression:
\[ |(3 - \sec^2 x)_{\text{max}} - (4 + \tan^2 y)_{\text{min}}| = |2 - 4| = |-2| = 2 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{2} \]