The value of sec `theta` can

Options

A. never be greater than 1
B. never be less than 1
C. never be equal to 1
D. never lie between -1 and 1

Answer

To solve the question regarding the value of sec θ, we need to analyze the properties of the secant function in trigonometry.

### Step-by-Step Solution:

1. **Understanding Secant Function**:
The secant function is defined as the reciprocal of the cosine function:
\[ \sec \theta = \frac{1}{\cos \theta} \]

2. **Range of Cosine Function**:
The cosine function (cos θ) has a range of:
\[ -1 \leq \cos \theta \leq 1 \]

3. **Finding the Range of Secant Function**:
Since sec θ is the reciprocal of cos θ, we need to consider the values of cos θ:
- When \(\cos \theta = 1\), \(\sec \theta = 1\)
- When \(\cos \theta = -1\), \(\sec \theta = -1\)
- When \(\cos \theta\) approaches 0 (from the positive side), \(\sec \theta\) approaches \(+\infty\).
- When \(\cos \theta\) approaches 0 (from the negative side), \(\sec \theta\) approaches \(-\infty\).

Therefore, the secant function can take values in the following intervals:
\[ (-\infty, -1] \cup [1, +\infty) \]

4. **Conclusion on the Value of Sec θ**:
From the range we derived, we can conclude that:
- The value of sec θ can never be between -1 and 1 (exclusive).
- The value of sec θ can be equal to -1 or 1.
- The value of sec θ can be less than -1 or greater than 1.

5. **Evaluating the Options**:
Given the options:
- **Option 1**: The value of sec θ can never be greater than 1. (Incorrect)
- **Option 2**: The value of sec θ can never be less than 1. (Incorrect)
- **Option 3**: The value of sec θ can never be equal to 1. (Incorrect)
- **Option 4**: The value of sec θ can never lie between -1 and 1. (Correct)

Thus, the correct answer is that the value of sec θ can never lie between -1 and 1.