Let $g(x) = ax + b$, where $a < 0$ and $g$ is defined from $[1, 3]$ onto $[0, 2]$. Then the value of $\cot(\cos^{-1}(|\sin x| + |\cos x|) + \sin^{-1}(-|\cos x| - |\sin x|))$ is equal to

Since $a < 0$, $g(x)$ is a strictly decreasing function.

Given $g: [1, 3] \rightarrow [0, 2]$ is an onto function, we have:

$g(1) = 2 \Rightarrow a + b = 2$

$g(3) = 0 \Rightarrow 3a + b = 0$

Solving these equations, we get $a = -1$ and $b = 3$.

Thus, $g(x) = -x + 3$.

Consider the expression $\cot(\cos^{-1}(|\sin x| + |\cos x|) + \sin^{-1}(-|\cos x| - |\sin x|))$.

Let $y = |\sin x| + |\cos x|$. We know that $1 \le |\sin x| + |\cos x| \le \sqrt{2}$ for all real $x$.

For $\cos^{-1}(y)$ and $\sin^{-1}(-y)$ to be defined, we must have $y \in [-1, 1]$ and $-y \in [-1, 1]$, which implies $y \le 1$.

The only possible value satisfying both conditions is $y = 1$.

Substituting $y = 1$ into the expression, we get:

$\cot(\cos^{-1}(1) + \sin^{-1}(-1)) = \cot\left(0 - \dfrac{\pi}{2}\right) = \cot\left(-\dfrac{\pi}{2}\right) = 0$.

Evaluating $g(x)$ at integer points:

$g(1) = 2$

$g(2) = 1$

$g(3) = 0$

Therefore, the value of the expression is equal to $g(3)$.

Answer: $g(3)$