The sum of all the elements in the range of \(f(x)=\operatorname{Sgn}(\sin x)+\operatorname{Sgn}(\cos x)+\operatorname{Sgn}(\tan x)+\operatorname{Sgn}(\cot x)\), \(\mathrm{x} \neq \frac{\mathrm{n} \pi}{2}, \mathrm{n} \in \mathbf{Z}\), where \(\operatorname{Sgn}(t)=\left\{\begin{array}{lll}1, & \text { if } & t>0 \\ -1 & \text { if } & t<0\end{array}\right.\), is

JEE Main · 2026 · 28 Jan 2026 · Shift 2 · Sets Relations and Functions

The sum of all the elements in the range of

f (x) = Sgn (sin x) + Sgn (cos x) + Sgn (tan x) + Sgn (cot x)

x \neq = \frac{n π}{2}, n \in Z

,
where

Sgn (t) = {1, - 1 if if t > 0 t < 0

, is

Answer: (B)

Away from the axes each sign term is constant on an open quadrant. For

x \in (0, \frac{π}{2})

all of

sin x, cos x, tan x, cot x

are positive, so

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

. For

x \in (\frac{π}{2}, π)

sin x > 0

while

cos x, tan x, cot x < 0

, so

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

. For

x \in (π, \frac{3 π}{2})

sin x, cos x < 0

and

tan x, cot x > 0

, so

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

. For

x \in (\frac{3 π}{2}, 2 π)

cos x > 0

and

sin x, tan x, cot x < 0

, so

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

. The range is

{- 2, 0, 4}

, and the sum of its elements is

- 2 + 0 + 4 = 2

JEE Main · 2026 · 28 Jan 2026 · Shift 2 · Single correct option