Question

The sum of all the elements in the range of
\[ f(x)=\operatorname{sgn}(\sin x)+\operatorname{sgn}(\cos x) +\operatorname{sgn}(\tan x)+\operatorname{sgn}(\cot x), \]
where
\[ x\neq \frac{n\pi}{2},\ n\in\mathbb{Z}, \]
and
\[ \operatorname{sgn}(t)= \begin{cases} 1, & t>0 \\ -1, & t<0 \end{cases} \]
is:

• A. \(0\)
• B. \(2\)
• C. \(-2\)
• D. \(4\)

Hint:

For sign-function problems involving trigonometric expressions, quadrant-wise analysis is the fastest and most reliable method.

Answer 1

The Correct Option is B

Concept: The sign of \(\sin x\), \(\cos x\), \(\tan x\), and \(\cot x\) depends on the quadrant in which the angle \(x\) lies. Since \(x\neq \frac{n\pi}{2}\), none of these trigonometric functions is zero. We evaluate \(f(x)\) separately in each quadrant.
Step 1: Quadrant-wise analysis Quadrant I \((0<x<\tfrac{\pi}{2})\): \[ \sin x>0,\ \cos x>0,\ \tan x>0,\ \cot x>0 \] \[ f(x)=1+1+1+1=4 \] Quadrant II \((\tfrac{\pi}{2}<x<\pi)\): \[ \sin x>0,\ \cos x<0,\ \tan x<0,\ \cot x<0 \] \[ f(x)=1-1-1-1=-2 \] Quadrant III \((\pi<x<\tfrac{3\pi}{2})\): \[ \sin x<0,\ \cos x<0,\ \tan x>0,\ \cot x>0 \] \[ f(x)=-1-1+1+1=0 \] Quadrant IV \((\tfrac{3\pi}{2}<x<2\pi)\): \[ \sin x<0,\ \cos x>0,\ \tan x<0,\ \cot x<0 \] \[ f(x)=-1+1-1-1=-2 \]
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 \[ 4+0+(-2)=2 \] Final Answer: \[ \boxed{2} \]