If `tan alpha = n tan beta and sin alpha = m sin beta`, then `cos^2 alpha` is

Options

A. `m^2/n^2`
B. `(m^2-1)/(n^2 -1)`
C. `(m^2 +1)/(n^2 +1)`
D. `m^2/(n^2 +1)`

Answer

To solve the problem, we need to find the value of \( \cos^2 \alpha \) given the equations \( \tan \alpha = n \tan \beta \) and \( \sin \alpha = m \sin \beta \).

### Step-by-step Solution:

1. **Express \( \tan \beta \) and \( \sin \beta \)**:
From the first equation, we can express \( \tan \beta \):
\[ \tan \beta = \frac{\tan \alpha}{n} \]
From the second equation, we can express \( \sin \beta \):
\[ \sin \beta = \frac{\sin \alpha}{m} \]

2. **Relate \( \cos \beta \) and \( \sin \beta \)**:
We know that \( \cos \beta = \frac{\sin \alpha}{m \tan \beta} \). Using the expression for \( \tan \beta \):
\[ \cos \beta = \frac{\sin \alpha}{m \left( \frac{\tan \alpha}{n} \right)} = \frac{n \sin \alpha}{m \tan \alpha} \]

3. **Use the identity \( \cos^2 \beta + \sin^2 \beta = 1 \)**:
We can express \( \cos^2 \beta \) in terms of \( \sin \alpha \) and \( \tan \alpha \):
\[ \cos^2 \beta = 1 - \sin^2 \beta = 1 - \left( \frac{\sin \alpha}{m} \right)^2 = 1 - \frac{\sin^2 \alpha}{m^2} \]

4. **Substituting \( \tan \beta \)**:
We can also express \( \tan^2 \beta \):
\[ \tan^2 \beta = \frac{\sin^2 \beta}{\cos^2 \beta} = \frac{\left( \frac{\sin \alpha}{m} \right)^2}{\cos^2 \beta} \]
From the first equation, substituting \( \tan \beta \):
\[ \tan^2 \beta = \frac{\tan^2 \alpha}{n^2} \]

5. **Relate \( \sin^2 \alpha \) and \( \cos^2 \alpha \)**:
Using the identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \):
\[ \cos^2 \alpha = 1 - \sin^2 \alpha \]

6. **Final expression for \( \cos^2 \alpha \)**:
We can relate everything back to \( \cos^2 \alpha \):
\[ \cos^2 \alpha = 1 - \frac{m^2 \tan^2 \alpha}{n^2} \]

7. **Substituting back to find \( \cos^2 \alpha \)**:
Since \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \), we can substitute and simplify to find:
\[ \cos^2 \alpha = \frac{m^2 - n^2}{m^2} \]

### Conclusion:
Thus, the final expression for \( \cos^2 \alpha \) is:
\[ \cos^2 \alpha = \frac{m^2 - n^2}{m^2} \]