Question

For $\theta > \frac{\pi}{3}$, the value of $f(\theta)=\sec ^{2} \theta+\cos ^{2} \theta$ always lies in the

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

Answer 1

Answer: (D)

We have, $f(\theta)=\sec ^{2} \theta+\cos ^{2} \theta$
Since AM $\geq$ GM
$\Rightarrow \frac{\sec ^{2} \theta+\cos ^{2} \theta}{2} \geq\left(\sec ^{2} \theta \cdot \cos ^{2} \theta\right)^{1 / 2}$
$\Rightarrow \frac{\sec ^{2} \theta+\cos ^{2} \theta}{2} \geq 1$
$\Rightarrow \frac{\sec ^{2} \theta+\cos ^{2} \theta}{2} \geq 2$
$\Rightarrow f(\theta) \geq 2$
$\therefore $ The value of $f(\theta)$ always lies in the interval $[2, \infty)$.