If `sin(theta-x) = a, cos (theta-y) = b`, then `cos(x-y)` may be equal to :

Options

A. `bsqrt((1-a^(2))) +asqrt((1-b^(2)))`
B. ab
C. `asqrt((1-b^(2))) -bsqrt((1-a^(2)))`
D. 2ab

Answer

To solve the problem where we need to find the value of \( \cos(x-y) \) given that \( \sin(\theta - x) = a \) and \( \cos(\theta - y) = b \), we can follow these steps:

### Step-by-Step Solution:

1. **Express \( \theta \) in terms of \( x \) and \( a \)**:
From the equation \( \sin(\theta - x) = a \), we can write:
\[ \theta - x = \sin^{-1}(a) \]
Therefore, we can express \( \theta \) as:
\[ \theta = x + \sin^{-1}(a) \tag{1} \]

2. **Express \( \theta \) in terms of \( y \) and \( b \)**:
From the equation \( \cos(\theta - y) = b \), we can write:
\[ \theta - y = \cos^{-1}(b) \]
Thus, we can express \( \theta \) as:
\[ \theta = y + \cos^{-1}(b) \tag{2} \]

3. **Set the two expressions for \( \theta \) equal to each other**:
From equations (1) and (2), we have:
\[ x + \sin^{-1}(a) = y + \cos^{-1}(b) \]

4. **Rearrange to find \( x - y \)**:
Rearranging the above equation gives:
\[ x - y = \cos^{-1}(b) - \sin^{-1}(a) \tag{3} \]

5. **Find \( \cos(x - y) \)**:
Now we need to find \( \cos(x - y) \). Using equation (3):
\[ \cos(x - y) = \cos(\cos^{-1}(b) - \sin^{-1}(a)) \]

6. **Apply the cosine difference formula**:
Using the cosine difference identity:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
where \( A = \cos^{-1}(b) \) and \( B = \sin^{-1}(a) \):
\[ \cos(x - y) = \cos(\cos^{-1}(b)) \cos(\sin^{-1}(a)) + \sin(\cos^{-1}(b)) \sin(\sin^{-1}(a)) \]

7. **Simplify using trigonometric identities**:
We know:
\[ \cos(\cos^{-1}(b)) = b \]
\[ \sin(\sin^{-1}(a)) = a \]
For \( \cos(\sin^{-1}(a)) \), we can use the identity \( \sin^2 + \cos^2 = 1 \):
\[ \cos(\sin^{-1}(a)) = \sqrt{1 - a^2} \]
For \( \sin(\cos^{-1}(b)) \):
\[ \sin(\cos^{-1}(b)) = \sqrt{1 - b^2} \]

8. **Combine the results**:
Substituting these into our equation gives:
\[ \cos(x - y) = b \cdot \sqrt{1 - a^2} + a \cdot \sqrt{1 - b^2} \]

### Final Result:
Thus, the value of \( \cos(x - y) \) is:
\[ \cos(x - y) = b \sqrt{1 - a^2} + a \sqrt{1 - b^2} \]