Let `g(x)= ax + b ` , where ` a lt 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 | - |sinx|)) ` is equal to :

Options

A. `g(1)`
B. `g(2)`
C. `g(3)`
D. `g(1) + g(3) `

Answer

To solve the given problem, we will break it down step by step.

### Step 1: Understanding the Function g(x)
We are given that \( g(x) = ax + b \) where \( a < 0 \) and \( g \) is defined from the interval \([1, 3]\) onto \([0, 2]\).

### Step 2: Setting Up the Equations
From the information provided:
1. \( g(1) = 2 \)
2. \( g(3) = 0 \)

Using these conditions, we can set up the following equations:
- For \( g(1) = 2 \):
\[ a(1) + b = 2 \quad \text{(Equation 1)} \]
- For \( g(3) = 0 \):
\[ a(3) + b = 0 \quad \text{(Equation 2)} \]

### Step 3: Solving the Equations
Now we have two equations:
1. \( a + b = 2 \)
2. \( 3a + b = 0 \)

We can subtract Equation 1 from Equation 2:
\[ (3a + b) - (a + b) = 0 - 2 \]
This simplifies to:
\[ 2a = -2 \implies a = -1 \]

Now we substitute \( a = -1 \) back into Equation 1:
\[ -1 + b = 2 \implies b = 3 \]

### Step 4: Formulating g(x)
Now we have determined the values of \( a \) and \( b \):
\[ g(x) = -x + 3 \]

### Step 5: Evaluating the Given Expression
Next, we need to evaluate:
\[ \cot\left(\cos^{-1}(|\sin x| + |\cos x|) + \sin^{-1}(-|\cos x| - |\sin x|)\right) \]

### Step 6: Finding the Range of \( |\sin x| + |\cos x| \)
The maximum value of \( |\sin x| + |\cos x| \) occurs when \( |\sin x| = |\cos x| = \frac{1}{\sqrt{2}} \):
\[ |\sin x| + |\cos x| \leq \sqrt{2} \]
The minimum value occurs when either \( |\sin x| \) or \( |\cos x| \) is 0, giving a minimum of 1.

### Step 7: Evaluating the Inverse Functions
1. The maximum value of \( |\sin x| + |\cos x| \) is \( \sqrt{2} \) and the minimum is \( 1 \).
2. Therefore, \( \cos^{-1}(|\sin x| + |\cos x|) \) will range from \( \cos^{-1}(1) = 0 \) to \( \cos^{-1}(\sqrt{2}) \) (not applicable since \( \sqrt{2} > 1 \)).
3. The term \( \sin^{-1}(-|\cos x| - |\sin x|) \) will be evaluated based on the range of \( |\sin x| + |\cos x| \).

### Step 8: Final Calculation
We take the minimum value for simplicity:
\[ \cot\left(0 + \sin^{-1}(-1)\right) = \cot\left(-\frac{\pi}{2}\right) = 0 \]

### Conclusion
Thus, the value of the expression is:
\[ \boxed{0} \]