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