If `x=sec theta-tan theta and y="cosec"theta+cot theta," then " y-x-xy=`

Answer

To solve the problem, we need to find the value of \( y - x - xy \) given that \( x = \sec \theta - \tan \theta \) and \( y = \csc \theta + \cot \theta \).

### Step-by-Step Solution:

1. **Substituting the values of x and y**:
\[ x = \sec \theta - \tan \theta \]
\[ y = \csc \theta + \cot \theta \]

2. **Expressing x and y in terms of sine and cosine**:
- Recall that:
\[ \sec \theta = \frac{1}{\cos \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Thus,
\[ x = \frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta} = \frac{1 - \sin \theta}{\cos \theta} \]

- For \( y \):
\[ \csc \theta = \frac{1}{\sin \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Therefore,
\[ y = \frac{1}{\sin \theta} + \frac{\cos \theta}{\sin \theta} = \frac{1 + \cos \theta}{\sin \theta} \]

3. **Finding \( y - x - xy \)**:
\[ y - x - xy = \left(\frac{1 + \cos \theta}{\sin \theta}\right) - \left(\frac{1 - \sin \theta}{\cos \theta}\right) - \left(\frac{1 + \cos \theta}{\sin \theta} \cdot \frac{1 - \sin \theta}{\cos \theta}\right) \]

4. **Finding the common denominator**:
The common denominator for the first two terms is \( \sin \theta \cos \theta \):
\[ y - x = \frac{(1 + \cos \theta) \cos \theta - (1 - \sin \theta) \sin \theta}{\sin \theta \cos \theta} \]

5. **Expanding the numerator**:
\[ = \frac{\cos \theta + \cos^2 \theta - \sin \theta + \sin^2 \theta}{\sin \theta \cos \theta} \]
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ = \frac{1 + \cos \theta - \sin \theta}{\sin \theta \cos \theta} \]

6. **Calculating \( xy \)**:
\[ xy = \left(\frac{1 + \cos \theta}{\sin \theta}\right) \left(\frac{1 - \sin \theta}{\cos \theta}\right) = \frac{(1 + \cos \theta)(1 - \sin \theta)}{\sin \theta \cos \theta} \]
Expanding:
\[ = \frac{1 - \sin \theta + \cos \theta - \sin \theta \cos \theta}{\sin \theta \cos \theta} \]

7. **Combining everything**:
Now, substituting back into \( y - x - xy \):
\[ y - x - xy = \frac{1 + \cos \theta - \sin \theta - (1 - \sin \theta + \cos \theta - \sin \theta \cos \theta)}{\sin \theta \cos \theta} \]
Simplifying:
\[ = \frac{1 + \cos \theta - \sin \theta - 1 + \sin \theta - \cos \theta + \sin \theta \cos \theta}{\sin \theta \cos \theta} \]
\[ = \frac{\sin \theta \cos \theta}{\sin \theta \cos \theta} = 1 \]

### Final Answer:
Thus, the value of \( y - x - xy \) is \( \boxed{1} \).