Q. The sum of all the elements in the range of

$ f(x)=\operatorname{Sgn}(\sin x)+\operatorname{Sgn}(\cos x)+\operatorname{Sgn}(\tan x)+\operatorname{Sgn}(\cot x),\ x\neq \frac{n\pi}{2},\ n\in\mathbb{Z}, $

where

$ \operatorname{Sgn}(t)= \begin{cases} 1, & \text{if } t>0,\\ -1, & \text{if } t<0, \end{cases} $

is :

Correct Answer: 2

Explanation

The values of $\sin x$, $\cos x$, $\tan x$ and $\cot x$ depend on the quadrant in which $x$ lies.

The domain excludes $x=\dfrac{n\pi}{2}$, so $\tan x$ and $\cot x$ are always defined.

Check the signs of all four functions in different quadrants.

---

First Quadrant $(0

$\sin x>0,\ \cos x>0,\ \tan x>0,\ \cot x>0$

$f(x)=1+1+1+1=4$

---

Second Quadrant $(\pi/2

$\sin x>0,\ \cos x<0,\ \tan x<0,\ \cot x<0$

$f(x)=1-1-1-1=-2$

---

Third Quadrant $(\pi

$\sin x<0,\ \cos x<0,\ \tan x>0,\ \cot x>0$

$f(x)=-1-1+1+1=0$

---

Fourth Quadrant $(3\pi/2

$\sin x<0,\ \cos x>0,\ \tan x<0,\ \cot x<0$

$f(x)=-1+1-1-1=-2$

---

Hence, the range of $f(x)$ is

$\{4,\,0,\,-2\}$

The sum of all the elements in the range is

$4+0+(-2)=2$

Therefore, the required sum is

2

Related JEE Main Questions