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 problem step by step, we will break it down into manageable parts:

### Step 1: Understanding the function \( g(x) = ax + b \)

Given that \( g(x) \) is defined from the interval \([1, 3]\) onto \([0, 2]\) and that \( a < 0 \), we can infer the following:

- The function is a linear function with a negative slope, which means it decreases as \( x \) increases.
- The maximum value of \( g(x) \) occurs at \( x = 1 \) and the minimum value occurs at \( x = 3 \).

### Step 2: Finding the values of \( g(1) \) and \( g(3) \)

From the problem statement:
- \( g(1) = 2 \) (maximum value)
- \( g(3) = 0 \) (minimum value)

### Step 3: Setting up equations

Using the values we have:
1. \( g(1) = a(1) + b = 2 \) → \( a + b = 2 \) (Equation 1)
2. \( g(3) = a(3) + b = 0 \) → \( 3a + b = 0 \) (Equation 2)

### Step 4: Solving the equations

Now, we can solve these two equations:

From Equation 2:
\[ b = -3a \]

Substituting \( b \) into Equation 1:
\[ a - 3a = 2 \]
\[ -2a = 2 \]
\[ a = -1 \]

Now substituting \( a \) back into the equation for \( b \):
\[ b = -3(-1) = 3 \]

Thus, we have:
\[ a = -1 \]
\[ b = 3 \]

### Step 5: Writing the function

Now, substituting \( a \) and \( b \) back into the function:
\[ g(x) = -x + 3 \]

### Step 6: Evaluating the expression

Now we need to evaluate:
\[ \cot\left(\cos^{-1}(|\sin x| + |\cos x|) + \sin^{-1}(-|\cos x| - |\sin x|)\right) \]

### Step 7: Finding the range of \( |\sin x| + |\cos x| \)

The maximum value of \( |\sin x| + |\cos x| \) occurs when \( |\sin x| = |\cos x| \), which is \( \sqrt{2} \). The minimum value occurs when either \( |\sin x| = 0 \) or \( |\cos x| = 0 \), which is \( 1 \).

Thus, the range of \( |\sin x| + |\cos x| \) is \( [1, \sqrt{2}] \).

### Step 8: Evaluating \( \cos^{-1} \) and \( \sin^{-1} \)

Using the maximum and minimum values:
1. For \( |\sin x| + |\cos x| = 1 \):
- \( \cos^{-1}(1) = 0 \)
- \( \sin^{-1}(-1) = -\frac{\pi}{2} \)
- Total: \( 0 - \frac{\pi}{2} = -\frac{\pi}{2} \)

2. For \( |\sin x| + |\cos x| = \sqrt{2} \):
- \( \cos^{-1}(\sqrt{2}) \) is not valid since it exceeds the range of cosine.

### Step 9: Final evaluation of \( \cot \)

Thus, we find:
\[ \cot(0 - \frac{\pi}{2}) = \cot(-\frac{\pi}{2}) = 0 \]

### Conclusion

The value of the expression is:
\[ \boxed{0} \]