If `alpha+beta=90^(@)andalpha:beta=2:1`, then the ratio of `cosalpha" to "cosbeta` is :

Options

A. `1:sqrt(3)`
B. `1:3`
C. `1:sqrt(2)`
D. `1 : 2`

Answer

To solve the problem, we need to find the ratio of \( \cos \alpha \) to \( \cos \beta \) given that \( \alpha + \beta = 90^\circ \) and the ratio \( \alpha : \beta = 2 : 1 \).

### Step-by-Step Solution:

1. **Set up the equations based on the given ratios**:
Since \( \alpha : \beta = 2 : 1 \), we can express \( \alpha \) in terms of \( \beta \):
\[ \alpha = 2\beta \]

2. **Use the sum of angles**:
We know that \( \alpha + \beta = 90^\circ \). Substituting \( \alpha = 2\beta \) into this equation gives:
\[ 2\beta + \beta = 90^\circ \]
Simplifying this, we have:
\[ 3\beta = 90^\circ \]

3. **Solve for \( \beta \)**:
Dividing both sides by 3:
\[ \beta = \frac{90^\circ}{3} = 30^\circ \]

4. **Find \( \alpha \)**:
Now substituting \( \beta \) back to find \( \alpha \):
\[ \alpha = 2\beta = 2 \times 30^\circ = 60^\circ \]

5. **Calculate \( \cos \alpha \) and \( \cos \beta \)**:
Now we find \( \cos \alpha \) and \( \cos \beta \):
\[ \cos \alpha = \cos 60^\circ = \frac{1}{2} \]
\[ \cos \beta = \cos 30^\circ = \frac{\sqrt{3}}{2} \]

6. **Find the ratio \( \frac{\cos \alpha}{\cos \beta} \)**:
Now we can find the ratio:
\[ \frac{\cos \alpha}{\cos \beta} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

7. **Express the ratio in simplest form**:
This can also be expressed as:
\[ \frac{\cos \alpha}{\cos \beta} = 1 : \sqrt{3} \]

### Final Answer:
The ratio of \( \cos \alpha \) to \( \cos \beta \) is \( 1 : \sqrt{3} \).