Question

The minimum value of the function $f\left(x\right)=\frac{\sin x}{\sqrt{1-{\cos }^{2}x}}+\frac{\cos x}{\sqrt{1-{\sin }^{2}x}}+\frac{\tan x}{\sqrt{{\sec }^{2}x}-1}+\frac{\cot x}{\sqrt{cose{c}^{2}x}-1}$ whenever it is defined is

• A. $4$
• B. $-2$
• C. $0$
• D. $2$

Answer 1

Answer: (B)

$-2$

The expression looks like $\frac{\sin x}{|\sin x|}+\frac{\cos x}{|\cos x|}+\frac{\tan x}{|\tan x|}+\frac{\cot x}{|\cot x|}$
The minimum value will be achieved when three of the terms become $-1$ and one term remains $1$.
Thus, either of sine or cosine will be positive and the rest three will be negative and thus the minimum value comes out as $-3+1=-2$