If `secx=sqrt2` and `(3pi)/(2)lt x lt 2pi` find the value of `(1-tanx-cosecx)/(1-cotx-cosecx)`

Answer

To solve the problem, we need to find the value of the expression \((1 - \tan x - \csc x) / (1 - \cot x - \csc x)\) given that \(\sec x = \sqrt{2}\) and \( \frac{3\pi}{2} < x < 2\pi\).

### Step-by-step Solution:

1. **Identify the Quadrant**:
Since \(x\) is in the interval \(\left(\frac{3\pi}{2}, 2\pi\right)\), we know that \(x\) is in the fourth quadrant.

2. **Find \(\cos x\)**:
Given \(\sec x = \sqrt{2}\), we can find \(\cos x\):
\[ \sec x = \frac{1}{\cos x} \implies \cos x = \frac{1}{\sec x} = \frac{1}{\sqrt{2}}. \]

3. **Find \(\sin x\)**:
Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\):
\[ \sin^2 x = 1 - \cos^2 x = 1 - \left(\frac{1}{\sqrt{2}}\right)^2 = 1 - \frac{1}{2} = \frac{1}{2}. \]
Therefore, \(\sin x = -\frac{1}{\sqrt{2}}\) (negative in the fourth quadrant).

4. **Calculate \(\tan x\)**:
\[ \tan x = \frac{\sin x}{\cos x} = \frac{-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = -1. \]

5. **Calculate \(\csc x\)**:
\[ \csc x = \frac{1}{\sin x} = -\sqrt{2}. \]

6. **Calculate \(\cot x\)**:
\[ \cot x = \frac{1}{\tan x} = \frac{1}{-1} = -1. \]

7. **Substitute Values into the Expression**:
Now substitute the values into the expression:
\[ \frac{1 - \tan x - \csc x}{1 - \cot x - \csc x} = \frac{1 - (-1) - (-\sqrt{2})}{1 - (-1) - (-\sqrt{2})}. \]

8. **Simplify the Numerator**:
\[ \text{Numerator: } 1 + 1 + \sqrt{2} = 2 + \sqrt{2}. \]

9. **Simplify the Denominator**:
\[ \text{Denominator: } 1 + 1 + \sqrt{2} = 2 + \sqrt{2}. \]

10. **Final Calculation**:
\[ \frac{2 + \sqrt{2}}{2 + \sqrt{2}} = 1. \]

### Final Answer:
The value of the expression \((1 - \tan x - \csc x) / (1 - \cot x - \csc x)\) is \(1\).