If `tan theta =-(1)/(sqrt10)and theta` lies in the fourth quadrant, then `sec theta =`

Options

A. `(1)/(sqrt11)`
B. `(-1)/(sqrt11)`
C. `sqrt((11)/(10))`
D. `-sqrt((10)/(11))`

Answer

To solve the problem, we need to find the value of \( \sec \theta \) given that \( \tan \theta = -\frac{1}{\sqrt{10}} \) and that \( \theta \) lies in the fourth quadrant.

### Step-by-Step Solution:

1. **Identify the given values:**
We know that:
\[ \tan \theta = -\frac{1}{\sqrt{10}} \]
Since \( \theta \) is in the fourth quadrant, \( \tan \theta \) is negative.

2. **Use the Pythagorean identity:**
We can use the identity:
\[ \tan^2 \theta + 1 = \sec^2 \theta \]
Substituting the value of \( \tan \theta \):
\[ \left(-\frac{1}{\sqrt{10}}\right)^2 + 1 = \sec^2 \theta \]

3. **Calculate \( \tan^2 \theta \):**
\[ \tan^2 \theta = \left(-\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} \]

4. **Substitute into the identity:**
\[ \frac{1}{10} + 1 = \sec^2 \theta \]

5. **Convert 1 to a fraction:**
\[ 1 = \frac{10}{10} \]
So,
\[ \sec^2 \theta = \frac{1}{10} + \frac{10}{10} = \frac{11}{10} \]

6. **Take the square root:**
\[ \sec \theta = \sqrt{\frac{11}{10}} \]

7. **Simplify the expression:**
\[ \sec \theta = \frac{\sqrt{11}}{\sqrt{10}} = \frac{\sqrt{11}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{11 \cdot 10}}{10} = \frac{\sqrt{110}}{10} \]
However, since we are interested in the secant value in the fourth quadrant, we take the positive root:
\[ \sec \theta = \frac{\sqrt{11}}{\sqrt{10}} = \frac{\sqrt{11}}{10} \]

### Final Answer:
\[ \sec \theta = \frac{\sqrt{11}}{\sqrt{10}} = \frac{\sqrt{11}}{10} \]