Question

If$-\frac{\pi }{2}<\theta <\frac{\pi }{2}$ ,then the minimum value of ${\cos }^3\theta +{\sec }^3\theta$ is

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

Answer 1

The correct option is B 2

Let $co{s}^3\theta$ and $se{c}^3\theta$ be two numbers. In the given interval, $cos\theta$ and $sec\theta$ are positive.
$AM\ge GM$
$\Rightarrow \frac{co{s}^3\theta +se{c}^3\theta }{2}\ge \sqrt{co{s}^3\theta se{c}^3\theta }$
$\Rightarrow co{s}^3\theta +se{c}^3\theta \ge 2$
Hence, option B is correct.