The value of `cos[2 tan^(-1)(-7)]` is

Options

A. `49/50`
B. `-49/50`
C. `24/25`
D. `-24/25`

Explanation

To find the value of \( \cos[2 \tan^{-1}(-7)] \), we can use the identity for \( \cos(2 \tan^{-1}(x)) \):

\[ \cos(2 \tan^{-1}(x)) = \frac{1 - x^2}{1 + x^2} \]

### Step-by-Step Solution:

1. **Identify the value of \( x \)**:
Here, \( x = -7 \).

2. **Calculate \( x^2 \)**:
\[ x^2 = (-7)^2 = 49 \]

3. **Substitute \( x \) into the identity**:
\[ \cos(2 \tan^{-1}(-7)) = \frac{1 - (-7)^2}{1 + (-7)^2} = \frac{1 - 49}{1 + 49} \]

4. **Simplify the expression**:
\[ = \frac{1 - 49}{1 + 49} = \frac{-48}{50} \]

5. **Further simplify**:
\[ = \frac{-24}{25} \]

Thus, the value of \( \cos[2 \tan^{-1}(-7)] \) is \( \frac{-24}{25} \).

### Final Answer:
\[ \cos[2 \tan^{-1}(-7)] = \frac{-24}{25} \]