If `sin^(-1)x+sin^(-1)y=(2pi)/(3), cos^(-1)x-cos^(-1)y=(pi)/(3)` then the number of values of (x, y) is :

Options

A. two
B. four
C. zero
D. None of these

Answer

To solve the problem, we have the following equations:

1. \( \sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3} \) (Equation 1)
2. \( \cos^{-1} x - \cos^{-1} y = \frac{\pi}{3} \) (Equation 2)

We need to find the number of values of \( (x, y) \).

### Step 1: Rewrite the second equation
Using the identity \( \cos^{-1} a = \frac{\pi}{2} - \sin^{-1} a \), we can rewrite Equation 2:

\[ \cos^{-1} x - \cos^{-1} y = \frac{\pi}{3} \implies \left(\frac{\pi}{2} - \sin^{-1} x\right) - \left(\frac{\pi}{2} - \sin^{-1} y\right) = \frac{\pi}{3} \]

This simplifies to:

\[ -\sin^{-1} x + \sin^{-1} y = \frac{\pi}{3} \]

Rearranging gives us:

\[ \sin^{-1} y - \sin^{-1} x = \frac{\pi}{3} \quad \text{(Equation 3)} \]

### Step 2: Add Equations 1 and 3
Now, we can add Equation 1 and Equation 3:

\[ (\sin^{-1} x + \sin^{-1} y) + (\sin^{-1} y - \sin^{-1} x) = \frac{2\pi}{3} + \frac{\pi}{3} \]

This simplifies to:

\[ 2\sin^{-1} y = \pi \]

Dividing both sides by 2 gives:

\[ \sin^{-1} y = \frac{\pi}{2} \]

### Step 3: Solve for \( y \)
From \( \sin^{-1} y = \frac{\pi}{2} \), we find:

\[ y = \sin\left(\frac{\pi}{2}\right) = 1 \]

### Step 4: Substitute \( y \) back into Equation 1
Now we substitute \( y = 1 \) back into Equation 1 to find \( x \):

\[ \sin^{-1} x + \sin^{-1}(1) = \frac{2\pi}{3} \]

Since \( \sin^{-1}(1) = \frac{\pi}{2} \), we have:

\[ \sin^{-1} x + \frac{\pi}{2} = \frac{2\pi}{3} \]

Subtracting \( \frac{\pi}{2} \) from both sides gives:

\[ \sin^{-1} x = \frac{2\pi}{3} - \frac{\pi}{2} \]

To simplify, we convert \( \frac{\pi}{2} \) to a common denominator:

\[ \sin^{-1} x = \frac{4\pi}{6} - \frac{3\pi}{6} = \frac{\pi}{6} \]

### Step 5: Solve for \( x \)
Now we find \( x \):

\[ x = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

### Conclusion
The values we have found are:

\[ (x, y) = \left(\frac{1}{2}, 1\right) \]

Since there are no other combinations of \( x \) and \( y \) that satisfy both equations, the number of values of \( (x, y) \) is **1**.