If `0lt xlt(pi)/(2)andsecx= "cosecy"` then the value of `sin(x+y)` is :

Options

A. 0
B. 1
C. `(1)/(2)`
D. `(1)/(sqrt(3))`

Answer

To solve the problem, we need to find the value of \( \sin(x + y) \) given that \( \sec x = \csc y \) and \( 0 < x < \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
We start with the equation \( \sec x = \csc y \).

2. **Expressing in Terms of Sine and Cosine**:
Recall that:
\[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \csc y = \frac{1}{\sin y} \]
Thus, we can rewrite the equation as:
\[ \frac{1}{\cos x} = \frac{1}{\sin y} \]
This implies:
\[ \sin y = \cos x \]

3. **Using the Pythagorean Identity**:
Since \( \sin^2 y + \cos^2 y = 1 \), we can find \( \sin^2 y \) and \( \cos^2 y \):
\[ \sin^2 y = \cos^2 x \]
Therefore, we can express \( \sin^2 x \) as:
\[ \sin^2 x = 1 - \cos^2 x = 1 - \sin^2 y \]

4. **Finding \( \sin(x + y) \)**:
We can use the sine addition formula:
\[ \sin(x + y) = \sin x \cos y + \cos x \sin y \]
From our earlier steps, we know:
\[ \sin y = \cos x \]
Therefore, substituting this into the sine addition formula gives:
\[ \sin(x + y) = \sin x \cos y + \cos x \cos x \]

5. **Using the Identity for \( \cos y \)**:
We can find \( \cos y \) using the identity \( \cos^2 y + \sin^2 y = 1 \):
\[ \cos^2 y = 1 - \sin^2 y = 1 - \cos^2 x \]
Thus, \( \cos y = \sqrt{1 - \cos^2 x} = \sin x \).

6. **Final Calculation**:
Now substituting \( \cos y = \sin x \) back into the equation:
\[ \sin(x + y) = \sin x \cdot \sin x + \cos x \cdot \cos x = \sin^2 x + \cos^2 x \]
By the Pythagorean identity, we know:
\[ \sin^2 x + \cos^2 x = 1 \]
Therefore:
\[ \sin(x + y) = 1 \]

### Conclusion:
The value of \( \sin(x + y) \) is \( 1 \).