What is the value of ` sec alpha - tan alpha, ` . If ` sec alpha + tan alpha = sqrt""3 ` ?

Options

A. `(1)/( sqrt(3))`
B. 0
C. 1
D. `sqrt""3`

Answer

To solve the problem, we need to find the value of \( \sec \alpha - \tan \alpha \) given that \( \sec \alpha + \tan \alpha = \sqrt{3} \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \sec \alpha + \tan \alpha = \sqrt{3} \]

2. **Use the identity for the difference of squares:**
We know that:
\[ \sec^2 \alpha - \tan^2 \alpha = 1 \]
This can be factored as:
\[ (\sec \alpha + \tan \alpha)(\sec \alpha - \tan \alpha) = 1 \]

3. **Substitute the known value:**
Substitute \( \sec \alpha + \tan \alpha \) with \( \sqrt{3} \):
\[ \sqrt{3}(\sec \alpha - \tan \alpha) = 1 \]

4. **Solve for \( \sec \alpha - \tan \alpha \):**
To isolate \( \sec \alpha - \tan \alpha \), divide both sides by \( \sqrt{3} \):
\[ \sec \alpha - \tan \alpha = \frac{1}{\sqrt{3}} \]

5. **Final answer:**
Therefore, the value of \( \sec \alpha - \tan \alpha \) is:
\[ \sec \alpha - \tan \alpha = \frac{1}{\sqrt{3}} \]