JEE Mathematics 2026 Question with Solution

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

Find: The sum of all distinct elements in the range of $f(x)$.

Concept: The sign of $\sin x$, $\cos x$, $\tan x$, and $\cot x$ depends on the quadrant in which $x$ lies. Since $x\neq \frac{n\pi}{2}$, none of these trigonometric functions is zero.

Step 1: Quadrant-wise analysis

Quadrant I $\left(0<x<\tfrac{\pi}{2}\right)$:

$$\sin x>0,\ \cos x>0,\ \tan x>0,\ \cot x>0$$ $$f(x)=1+1+1+1=4$$

Quadrant II $\left(\tfrac{\pi}{2}<x<\pi\right)$:

$$\sin x>0,\ \cos x<0,\ \tan x<0,\ \cot x<0$$ $$f(x)=1-1-1-1=-2$$

Quadrant III $\left(\pi<x<\tfrac{3\pi}{2}\right)$:

$$\sin x<0,\ \cos x<0,\ \tan x>0,\ \cot x>0$$ $$f(x)=-1-1+1+1=0$$

Quadrant IV $\left(\tfrac{3\pi}{2}<x<2\pi\right)$:

$$\sin x<0,\ \cos x>0,\ \tan x<0,\ \cot x<0$$ $$f(x)=-1+1-1-1=-2$$

Step 2: Determine the range From all quadrants, the distinct values taken by $f(x)$ are:

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

Step 3: Sum of all elements in the range

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

Therefore, the sum of all elements in the range is $2$. The correct option is B.