The value of `cos ^(2) theta + sec ^(2) theta ` is always

Options

A. less than 1
B. equal to 1
C. greater then 1, but less than 2
D. greater than or equal 2

Answer

To solve the problem, we need to find the value of \( \cos^2 \theta + \sec^2 \theta \) and determine its range.

### Step-by-Step Solution:

1. **Recall the definition of secant**:
\[ \sec \theta = \frac{1}{\cos \theta} \]
Therefore,
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \]

2. **Substitute secant into the expression**:
\[ \cos^2 \theta + \sec^2 \theta = \cos^2 \theta + \frac{1}{\cos^2 \theta} \]

3. **Let \( x = \cos^2 \theta \)**, where \( x \) is in the range \( (0, 1] \) since \( \cos^2 \theta \) cannot be negative. The expression now becomes:
\[ x + \frac{1}{x} \]

4. **Find the minimum value of the expression**:
To find the minimum value of \( x + \frac{1}{x} \), we can use the AM-GM inequality:
\[ \frac{x + \frac{1}{x}}{2} \geq \sqrt{x \cdot \frac{1}{x}} = 1 \]
This implies:
\[ x + \frac{1}{x} \geq 2 \]

5. **Conclusion**:
Since \( x + \frac{1}{x} \) is always greater than or equal to 2, we conclude that:
\[ \cos^2 \theta + \sec^2 \theta \geq 2 \]

Thus, the value of \( \cos^2 \theta + \sec^2 \theta \) is always greater than or equal to 2.

### Final Answer:
The value of \( \cos^2 \theta + \sec^2 \theta \) is always greater than or equal to 2.

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