If `tanalpha=(1)/(7)&tanbeta=(1)/(3)`, then `cos2alpha=?`

Options

A. `sin2beta`
B. `sin4beta`
C. `sin3beta`
D. `cos3beta`

Answer

To find \( \cos 2\alpha \) given that \( \tan \alpha = \frac{1}{7} \) and \( \tan \beta = \frac{1}{3} \), we can follow these steps:

### Step 1: Calculate \( \tan 2\beta \)

We use the formula for \( \tan 2\beta \):
\[ \tan 2\beta = \frac{2 \tan \beta}{1 - \tan^2 \beta} \]

Substituting \( \tan \beta = \frac{1}{3} \):
\[ \tan 2\beta = \frac{2 \cdot \frac{1}{3}}{1 - \left(\frac{1}{3}\right)^2} \]
Calculating \( \tan^2 \beta \):
\[ \tan^2 \beta = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
Thus:
\[ 1 - \tan^2 \beta = 1 - \frac{1}{9} = \frac{8}{9} \]
Now substituting back:
\[ \tan 2\beta = \frac{\frac{2}{3}}{\frac{8}{9}} = \frac{2}{3} \cdot \frac{9}{8} = \frac{3}{4} \]

### Step 2: Relate \( \alpha \) and \( \beta \)

We use the formula for \( \tan(\alpha + 2\beta) \):
\[ \tan(\alpha + 2\beta) = \frac{\tan \alpha + \tan 2\beta}{1 - \tan \alpha \tan 2\beta} \]

Substituting \( \tan \alpha = \frac{1}{7} \) and \( \tan 2\beta = \frac{3}{4} \):
\[ \tan(\alpha + 2\beta) = \frac{\frac{1}{7} + \frac{3}{4}}{1 - \left(\frac{1}{7} \cdot \frac{3}{4}\right)} \]

### Step 3: Calculate the numerator

Finding a common denominator for the numerator:
\[ \frac{1}{7} + \frac{3}{4} = \frac{4}{28} + \frac{21}{28} = \frac{25}{28} \]

### Step 4: Calculate the denominator

Calculating the denominator:
\[ 1 - \left(\frac{1}{7} \cdot \frac{3}{4}\right) = 1 - \frac{3}{28} = \frac{28 - 3}{28} = \frac{25}{28} \]

### Step 5: Simplify \( \tan(\alpha + 2\beta) \)

Now substituting back:
\[ \tan(\alpha + 2\beta) = \frac{\frac{25}{28}}{\frac{25}{28}} = 1 \]

### Step 6: Find the angle

Since \( \tan(\alpha + 2\beta) = 1 \), we have:
\[ \alpha + 2\beta = \frac{\pi}{4} \]
This implies:
\[ \alpha = \frac{\pi}{4} - 2\beta \]

### Step 7: Find \( \cos 2\alpha \)

Now, multiplying the equation by 2:
\[ 2\alpha = \frac{\pi}{2} - 4\beta \]
Using the cosine identity:
\[ \cos 2\alpha = \cos\left(\frac{\pi}{2} - 4\beta\right) = \sin 4\beta \]

### Final Result

Thus, we conclude:
\[ \cos 2\alpha = \sin 4\beta \]