If `x=secphi-tanphi&y=cosecphi+cotphi`, then

Options

A. `x=(y+1)/(y-1)`
B. `y=(1-x)/(1+x)`
C. `x=(y-1)/(y+1)`
D. `y=(1+x)/(1-x)`

Answer

To solve the problem, we need to find the relationship between \( x \) and \( y \) given that:

\[ x = \sec \phi - \tan \phi \]
\[ y = \csc \phi + \cot \phi \]

We will evaluate \( x \) and \( y \) for a specific angle, \( \phi = 45^\circ \).

### Step 1: Calculate \( x \)

1. **Find \( \sec 45^\circ \)**:
\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

2. **Find \( \tan 45^\circ \)**:
\[ \tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = 1 \]

3. **Substitute into \( x \)**:
\[ x = \sec 45^\circ - \tan 45^\circ = \sqrt{2} - 1 \]

### Step 2: Calculate \( y \)

1. **Find \( \csc 45^\circ \)**:
\[ \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

2. **Find \( \cot 45^\circ \)**:
\[ \cot 45^\circ = \frac{\cos 45^\circ}{\sin 45^\circ} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = 1 \]

3. **Substitute into \( y \)**:
\[ y = \csc 45^\circ + \cot 45^\circ = \sqrt{2} + 1 \]

### Step 3: Analyze the relationship between \( x \) and \( y \)

Now we have:
- \( x = \sqrt{2} - 1 \)
- \( y = \sqrt{2} + 1 \)

### Step 4: Compare \( x \) and \( y \)

To find a relationship between \( x \) and \( y \), we can express \( y \) in terms of \( x \):

1. **Express \( y \)**:
\[ y = x + 2 \]

### Conclusion

Thus, the relationship between \( x \) and \( y \) is:
\[ y = x + 2 \]