Question

For \(\theta>\pi/3\) the value of \(f(\theta) = \sec^2\theta + \cos^2\theta\) always lies in the interval

• A. \((0, 2)\)
• B. \([0, 1]\)
• C. \((1, 2)\)
• D. \([2, \infty)\)

Hint:

For positive numbers, AM \(\ge\) GM, equality when numbers equal.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
AM \(\ge\) GM: \(\frac{a + b}{2} \ge \sqrt{ab}\). 
Step 2: Detailed Explanation: 
\(\sec^2\theta + \cos^2\theta \ge 2\sqrt{\sec^2\theta \cdot \cos^2\theta} = 2\) 
Equality when \(\sec^2\theta = \cos^2\theta \rightarrow \cos^4\theta = 1 \rightarrow \cos\theta = \pm 1 \rightarrow \theta = 0, \pi\) 
But \(\theta>\pi/3\), so minimum \(> 2\)? Actually \(\sec^2\theta + \cos^2\theta \ge 2\) always. 
Step 3: Final Answer: 
\([2, \infty)\).