Animated Solution for Mathematics - Trigonometry: The sum of all the elements in the range of $f(x)=\text{Sgn}(\sin x)+\text{Sgn}(\cos x)+\text{Sgn}(\tan x)+\text{Sgn}(\cot x)$, $x\ne \frac{n\pi }{2},n\in \mathbb{Z}$, where $\text{Sgn}(t)=\{\begin{array}{ll}1,& \text{if }t>0\\ -1,& \text{if }t<0\end{array}$ is :

The function provided is defined as:

The signum function, $\text{Sgn}(u)$, is defined as $1$ if $u>0$, $-1$ if $u<0$, and $0$ if $u=0$. To find the range of $f(x)$, we must evaluate the sum across the four quadrants of the unit circle, noting that the function is undefined where any of the trigonometric ratios are zero.

The Land of Positivity

In the first quadrant, where $x\in (0,\frac{\pi }{2})$, all trigonometric ratios are positive.

Applying the signum function to each term:

Thus, $4$ is the first element of our range.

The Sine Sanctuary

In the second quadrant, where $x\in (\frac{\pi }{2},\pi )$, only $\sin x$ is positive. Consequently, $\cos x$, $\tan x$, and $\cot x$ are negative.

The function evaluates as follows:

Thus, $-2$ is the second element of our range.

The Tangent Territory

In the third quadrant, where $x\in (\pi ,\frac{3\pi }{2})$, both $\tan x$ and $\cot x$ are positive, while $\sin x$ and $\cos x$ are negative.

The function evaluates as follows:

Thus, $0$ is the third element of our range.

The Cosine Kingdom

In the fourth quadrant, where $x\in (\frac{3\pi }{2},2\pi )$, only $\cos x$ is positive. The remaining functions, $\sin x$, $\tan x$, and $\cot x$, are negative.

The function evaluates as follows:

This confirms our previous result of $-2$.

We have traversed the entire circle and identified the set of unique values for the range of $f(x)$:

The problem asks for the sum of all elements in this range:

The final sum of the elements in the range is $2$.