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, we start with the given equations:

1. \( \sin(\theta - x) = a \)
2. \( \cos(\theta - y) = b \)

We need to find the value of \( \cos(x - y) \).

### Step 1: Express \( \theta \) in terms of \( x \) and \( a \)

From the first equation, we can express \( \theta \) as follows:

\[ \theta - x = \sin^{-1}(a) \implies \theta = x + \sin^{-1}(a) \]

### Step 2: Express \( \theta \) in terms of \( y \) and \( b \)

From the second equation, we can express \( \theta \) similarly:

\[ \theta - y = \cos^{-1}(b) \implies \theta = y + \cos^{-1}(b) \]

### Step 3: Set the two expressions for \( \theta \) equal to each other

Now we have two expressions for \( \theta \):

\[ x + \sin^{-1}(a) = y + \cos^{-1}(b) \]

### Step 4: Rearrange the equation to find \( x - y \)

Rearranging gives us:

\[ x - y = \cos^{-1}(b) - \sin^{-1}(a) \]

### Step 5: Find \( \cos(x - y) \)

Now we need to find \( \cos(x - y) \):

\[ \cos(x - y) = \cos(\cos^{-1}(b) - \sin^{-1}(a)) \]

### Step 6: Use the cosine subtraction formula

Using the cosine subtraction formula:

\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]

We can substitute \( a = \cos^{-1}(b) \) and \( b = \sin^{-1}(a) \):

\[ \cos(x - y) = \cos(\cos^{-1}(b)) \cos(\sin^{-1}(a)) + \sin(\cos^{-1}(b)) \sin(\sin^{-1}(a)) \]

### Step 7: Simplify the expression

Now we simplify each term:

1. \( \cos(\cos^{-1}(b)) = b \)
2. \( \sin(\sin^{-1}(a)) = a \)
3. \( \sin(\cos^{-1}(b)) = \sqrt{1 - b^2} \) (using the identity \( \sin^2 + \cos^2 = 1 \))
4. \( \cos(\sin^{-1}(a)) = \sqrt{1 - a^2} \)

Substituting these values into the equation gives:

\[ \cos(x - y) = b \cdot \sqrt{1 - a^2} + a \cdot \sqrt{1 - b^2} \]

### Final Answer

Thus, the value of \( \cos(x - y) \) is:

\[ \cos(x - y) = b \sqrt{1 - a^2} + a \sqrt{1 - b^2} \]