If `tanA=(1)/(2),tanB=(1)/(3)`, then `cos2A=?`

Options

A. sin B
B. sin 2B
C. sin 3B
D. cos 3B

Answer

To solve the problem, we need to find the value of \( \cos 2A \) given that \( \tan A = \frac{1}{2} \) and \( \tan B = \frac{1}{3} \).

### Step-by-Step Solution:

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

2. **Substitute the value of \( \tan A \)**:
Since \( \tan A = \frac{1}{2} \), we calculate \( \tan^2 A \):
\[ \tan^2 A = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
Now substitute this into the formula:
\[ \cos 2A = \frac{1 - \frac{1}{4}}{1 + \frac{1}{4}} \]

3. **Simplify the numerator and denominator**:
- For the numerator:
\[ 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \]
- For the denominator:
\[ 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4} \]

4. **Combine the results**:
\[ \cos 2A = \frac{\frac{3}{4}}{\frac{5}{4}} = \frac{3}{5} \]

Thus, the value of \( \cos 2A \) is \( \frac{3}{5} \).