The sum of all the elements in the range of f | vedclass

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Question

(B) We analyze the function $f(x)$ in different quadrants for $x 
eq \frac{n\pi}{2}$:$1$. For $x \in (0, \pi/2)$, $\sin x > 0, \cos x > 0, 	an x > 0

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$2$

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(B) We analyze the function $f(x)$ in different quadrants for $x 
eq \frac{n\pi}{2}$:$1$. For $x \in (0, \pi/2)$, $\sin x > 0, \cos x > 0, 	an x > 0, \cot x > 0 \Rightarrow f(x) = 1 + 1 + 1 + 1 = 4$.$2$. For $x \in (\pi/2, \pi)$, $\sin x > 0, \cos x $3$. For $x \in (\pi, 3\pi/2)$, $\sin x  0, \cot x > 0 \Rightarrow f(x) = -1 - 1 + 1 + 1 = 0$.$4$. For $x \in (3\pi/2, 2\pi)$, $\sin x  0, 	an x Thus, the range of $f(x)$ is $\{-2, 0, 4\}$.The sum of the elements in the range is $-2 + 0 + 4 = 2$.

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

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