If `secx=sqrt(2)` and x does not lie in the `1^(st)` quadrant, find the value of `(1+tanx+" cosec "x)/(1+cot x-" cosec "x)`.

Answer

To solve the problem step by step, we start with the given information:

**Given:**
- \( \sec x = \sqrt{2} \)
- \( x \) does not lie in the first quadrant.

### Step 1: Identify the quadrant
Since \( \sec x \) is positive and \( x \) does not lie in the first quadrant, it must be in the fourth quadrant.

### Step 2: Determine the angle
We know that:
\[ \sec 45^\circ = \sqrt{2} \]
Since \( x \) is in the fourth quadrant, we can express \( x \) as:
\[ x = 360^\circ - 45^\circ = 315^\circ \]

### Step 3: Calculate \( \tan x \) and \( \csc x \)
In the fourth quadrant:
- \( \tan(360^\circ - \theta) = -\tan \theta \)
- \( \csc(360^\circ - \theta) = -\csc \theta \)

Thus:
\[ \tan 315^\circ = -\tan 45^\circ = -1 \]
\[ \csc 315^\circ = -\csc 45^\circ = -\sqrt{2} \]

### Step 4: Substitute values into the expression
Now we substitute \( \tan x \) and \( \csc x \) into the expression:
\[ \frac{1 + \tan x + \csc x}{1 + \cot x - \csc x} \]
Substituting the values:
\[ \tan x = -1, \quad \csc x = -\sqrt{2}, \quad \cot x = \frac{1}{\tan x} = -1 \]
Thus, the expression becomes:
\[ \frac{1 - 1 - \sqrt{2}}{1 - 1 + \sqrt{2}} \]

### Step 5: Simplify the expression
The numerator simplifies to:
\[ 1 - 1 - \sqrt{2} = -\sqrt{2} \]
The denominator simplifies to:
\[ 1 - 1 + \sqrt{2} = \sqrt{2} \]
So we have:
\[ \frac{-\sqrt{2}}{\sqrt{2}} = -1 \]

### Final Answer
The value of the expression is:
\[ \boxed{-1} \]