If `tan A =(1)/(7)`, then what is cos2A equal to ?

Options

A. `(24)/(25)`
B. `(18)/(25)`
C. `(12)/(25)`
D. `(6)/(25)`

Answer

To find \( \cos 2A \) given that \( \tan A = \frac{1}{7} \), we can use the formula for \( \cos 2A \) in terms of \( \tan A \):

\[ \cos 2A = \frac{1 - \tan^2 A}{1 + \tan^2 A} \]

### Step-by-Step Solution:

1. **Identify the value of \( \tan A \)**:
\[ \tan A = \frac{1}{7} \]

2. **Calculate \( \tan^2 A \)**:
\[ \tan^2 A = \left(\frac{1}{7}\right)^2 = \frac{1}{49} \]

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

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

5. **Simplify the denominator**:
\[ 1 + \frac{1}{49} = \frac{49}{49} + \frac{1}{49} = \frac{50}{49} \]

6. **Combine the simplified numerator and denominator**:
\[ \cos 2A = \frac{\frac{48}{49}}{\frac{50}{49}} = \frac{48}{50} \]

7. **Further simplify the fraction**:
\[ \frac{48}{50} = \frac{24}{25} \]

### Final Answer:
\[ \cos 2A = \frac{24}{25} \]