Question

The range of function $f\left(x\right)=\frac{1}{\left|\sin x\right|}+\frac{1}{\left|\cos x\right|}$ is

• A. $\left(2\sqrt{2},\infty \right)$
• B. $(2,2\sqrt{2})$
• C. $(0,2\sqrt{2})$
• D. $(-\infty ,2\sqrt{2})$

Answer 1

The correct option is A $\left(2\sqrt{2},\infty \right)$

Let $f\left(x\right)=\frac{1}{\left|\sin x\right|}+\frac{1}{\left|\cos x\right|}$,

As we know that,

$AM\ge GM$

$\begin{array}{cc}\frac{1}{\left|\sin x\right|}+\frac{1}{\left|\cos x\right|}& \ge 2\sqrt{\frac{1}{\left|\sin x\right|\left|\cos x\right|}}\\ & =2\sqrt{2\cos ec\left(2x\right)}\end{array}$

$\Rightarrow \frac{1}{\left|\sin x\right|}+\frac{1}{\left|\cos x\right|}\ge 2\sqrt{2}$

$\therefore Range=\left(2\sqrt{2},\infty \right)$

Hence option $\left(A\right)$ is correct.