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

To solve this problem, we need to evaluate the function \(f(x) = \operatorname{sgn}(\sin x) + \operatorname{sgn}(\cos x) + \operatorname{sgn}(\tan x) + \operatorname{sgn}(\cot x)\) for all permissible values of \(x\) where \(x \neq \frac{n\pi}{2}\), \(n \in \mathbb{Z}\). Here's a step-by-step analysis:

1. First, consider the function \(\operatorname{sgn}(t)\) which is defined as: 

\(\operatorname{sgn}(t) = 1\)	if \(t > 0\)
\(\operatorname{sgn}(t) = -1\)	if \(t < 0\)

1. Evaluate the sign of each trigonometric function within specific intervals because the sign depends on the quadrant of the angle:
   • For \(0 < x < \frac{\pi}{2}\) (First Quadrant):
     • \(\sin x > 0\), \(\cos x > 0\), \(\tan x > 0\), \(\cot x > 0\)
     • Thus, \(f(x) = \operatorname{sgn}(\sin x) + \operatorname{sgn}(\cos x) + \operatorname{sgn}(\tan x) + \operatorname{sgn}(\cot x) = 1 + 1 + 1 + 1 = 4\)
   • For \(\frac{\pi}{2} < x < \pi\) (Second Quadrant):
     • \(\sin x > 0\), \(\cos x < 0\), \(\tan x < 0\), \(\cot x < 0\)
     • Thus, \(f(x) = 1 - 1 - 1 - 1 = -2\)
   • For \(\pi < x < \frac{3\pi}{2}\) (Third Quadrant):
     • \(\sin x < 0\), \(\cos x < 0\), \(\tan x > 0\), \(\cot x > 0\)
     • Thus, \(f(x) = -1 - 1 + 1 + 1 = 0\)
   • For \(\frac{3\pi}{2} < x < 2\pi\) (Fourth Quadrant):
     • \(\sin x < 0\), \(\cos x > 0\), \(\tan x < 0\), \(\cot x < 0\)
     • Thus, \(f(x) = -1 + 1 - 1 - 1 = -2\)
2. The range of \(f(x)\) within permissible \(x\) is \(\{4, -2, 0\}\).
3. Adding all the elements in this range: \(4 + (-2) + 0 = 2\).

Therefore, the sum of all elements in the range of the function is \(2\), which matches the correct answer \(2\).