If `tanalpha =(1)/(7) and tanbeta =(1)/(3) , then, cos2alpha` is equal to

Options

A. `sin2beta`
B. `sin4beta`
C. `sin2beta`
D. `cos2beta`

Explanation

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 will use the double angle formula for tangent:
\[ \tan 2\beta = \frac{2 \tan \beta}{1 - \tan^2 \beta} \]
Substituting \( \tan \beta = \frac{1}{3} \):
\[ \tan 2\beta = \frac{2 \cdot \frac{1}{3}}{1 - \left(\frac{1}{3}\right)^2} \]
Calculating the denominator:
\[ 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \]
Thus,
\[ \tan 2\beta = \frac{\frac{2}{3}}{\frac{8}{9}} = \frac{2}{3} \cdot \frac{9}{8} = \frac{3}{4} \]

### Step 2: Calculate \( \tan(\alpha + 2\beta) \)
Using the formula for the tangent of a sum:
\[ \tan(\alpha + 2\beta) = \frac{\tan \alpha + \tan 2\beta}{1 - \tan \alpha \tan 2\beta} \]
Substituting \( \tan \alpha = \frac{1}{7} \) and \( \tan 2\beta = \frac{3}{4} \):
\[ \tan(\alpha + 2\beta) = \frac{\frac{1}{7} + \frac{3}{4}}{1 - \left(\frac{1}{7} \cdot \frac{3}{4}\right)} \]
Finding a common denominator for the numerator:
\[ \frac{1}{7} + \frac{3}{4} = \frac{4}{28} + \frac{21}{28} = \frac{25}{28} \]
Calculating the denominator:
\[ 1 - \left(\frac{1}{7} \cdot \frac{3}{4}\right) = 1 - \frac{3}{28} = \frac{28 - 3}{28} = \frac{25}{28} \]
Thus,
\[ \tan(\alpha + 2\beta) = \frac{\frac{25}{28}}{\frac{25}{28}} = 1 \]

### Step 3: Relate \( \tan(\alpha + 2\beta) \) to angles
Since \( \tan(\alpha + 2\beta) = 1 \), we can express this as:
\[ \alpha + 2\beta = \frac{\pi}{4} \]

### Step 4: Solve for \( \alpha \)
Rearranging gives:
\[ \alpha = \frac{\pi}{4} - 2\beta \]

### Step 5: Find \( \cos 2\alpha \)
Using the cosine double angle identity:
\[ \cos 2\alpha = \cos\left(2\left(\frac{\pi}{4} - 2\beta\right)\right) = \cos\left(\frac{\pi}{2} - 4\beta\right) \]
Using the co-function identity:
\[ \cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta \]
Thus,
\[ \cos 2\alpha = \sin 4\beta \]

### Final Answer
Therefore, \( \cos 2\alpha = \sin 4\beta \).

---