If `x + y = 90^(@)` , then what is
`sqrt( cos x cosec y - cos x sin y ) ` equal to

Options

A. cos x
B. sin x
C. `sqrt(cos x)`
D. ` sqrt(sin x)`

Answer

To solve the problem, we need to evaluate the expression \( \sqrt{ \cos x \csc y - \cos x \sin y } \) given that \( x + y = 90^\circ \).

### Step-by-Step Solution:

1. **Substituting for \( y \)**:
Since \( x + y = 90^\circ \), we can express \( y \) as:
\[ y = 90^\circ - x \]

2. **Rewriting the Expression**:
Substitute \( y \) in the expression:
\[ \sqrt{ \cos x \csc(90^\circ - x) - \cos x \sin(90^\circ - x) } \]

3. **Using Trigonometric Identities**:
We know from trigonometric identities:
- \( \csc(90^\circ - x) = \sec x \)
- \( \sin(90^\circ - x) = \cos x \)

Therefore, we can rewrite the expression as:
\[ \sqrt{ \cos x \sec x - \cos x \cos x } \]

4. **Simplifying the Expression**:
We know that \( \sec x = \frac{1}{\cos x} \), so:
\[ \cos x \sec x = \cos x \cdot \frac{1}{\cos x} = 1 \]
Thus, the expression simplifies to:
\[ \sqrt{ 1 - \cos^2 x } \]

5. **Using the Pythagorean Identity**:
We know from the Pythagorean identity that:
\[ 1 - \cos^2 x = \sin^2 x \]
Therefore, we have:
\[ \sqrt{ \sin^2 x } \]

6. **Taking the Square Root**:
The square root of \( \sin^2 x \) gives us:
\[ |\sin x| \]
Since we are considering angles in the first quadrant (where sine is positive), we have:
\[ \sqrt{ \sin^2 x } = \sin x \]

### Final Answer:
Thus, the value of \( \sqrt{ \cos x \csc y - \cos x \sin y } \) is:
\[ \sin x \]