`If-(pi)/(2)lt0lt(pi)/(2),` then the minimum value of `cos^(3)theta+sec^(3)thets` is

Options

A. 1
B. 2
C. 0
D. none of these.

Explanation

To find the minimum value of \( \cos^3 \theta + \sec^3 \theta \) for \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Define the terms**: Let \( x = \cos^3 \theta \) and \( y = \sec^3 \theta \). We know that \( \sec \theta = \frac{1}{\cos \theta} \), so \( y = \sec^3 \theta = \frac{1}{\cos^3 \theta} = \frac{1}{x} \).

2. **Apply the AM-GM inequality**: According to the AM-GM inequality, for any non-negative numbers \( a \) and \( b \):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
In our case, we apply it to \( x \) and \( y \):
\[ \frac{\cos^3 \theta + \sec^3 \theta}{2} \geq \sqrt{\cos^3 \theta \cdot \sec^3 \theta} \]

3. **Calculate the product**: The product \( \cos^3 \theta \cdot \sec^3 \theta \) simplifies as follows:
\[ \cos^3 \theta \cdot \sec^3 \theta = \cos^3 \theta \cdot \frac{1}{\cos^3 \theta} = 1 \]

4. **Substitute back into the inequality**: Now, substituting back into the AM-GM inequality:
\[ \frac{\cos^3 \theta + \sec^3 \theta}{2} \geq \sqrt{1} = 1 \]
Therefore,
\[ \cos^3 \theta + \sec^3 \theta \geq 2 \]

5. **Conclusion**: The minimum value of \( \cos^3 \theta + \sec^3 \theta \) is 2. This minimum occurs when \( \cos^3 \theta = \sec^3 \theta \), which happens when \( \cos \theta = 1 \) (i.e., \( \theta = 0 \)).

### Final Answer:
The minimum value of \( \cos^3 \theta + \sec^3 \theta \) is \( 2 \).

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